Two new classification theorems on residuated monoids

Abstract

We present two very recent Mostert-Shields style classification theorems on residuated l-monoids along with some related results in sub structural logics.

Full text

LINZ
2012
33rd Linz Seminar on
Fuzzy Set Theory
Abstracts
Enriched Category Theory
and Related Topics
Bildungszentrum St. Magdalena, Linz, Austria
February 14 – 18, 2012
Abstracts
Ulrich Höhle
Lawrence N. Stout
Erich Peter Klement
Editors

LINZ 2012
—
ENRICHED CATEGORY THEORY
AND RELATED TOPICS
ABSTRACTS
Ulrich H¨ohle, Lawrence N. Stout, Erich Peter Klement
Editors
Printed by: Universit¨atsdirektion, Johannes Kepler Universit¨at, A-4040 Linz

2

Since their inception in 1979, the Linz Seminars on Fuzzy Set Theory have
emphasized the development of mathematical aspects of fuzzy sets by bring-
ing together researchers in fuzzy sets and established mathematicians whose
work outside of fuzzy set theory can provide directions for further research. The
philosophy of the seminar has always been to keep it deliberately small and
intimate so that informal critical discussions remain central. There are no paral-
lel sessions and during the week there are several round tables to discuss open
problems and promising directions for further work.
LINZ 2012 will be the 33rd seminar carrying on this tradition and is devoted
to the theme “Enriched Category Theory and Related Topics”. The goal of the
seminar is to present and to discuss recent advances in enriched category theory
and its various applications in pure and applied mathematics.
A large number of highly interesting contributions were submitted for pre-
sentation at LINZ 2012. This volume contains the abstracts of this impressive
collection. The regular contributions are complemented by four invited talks
which are intended to give new ideas and impulses from outside the traditional
Linz Seminar community.
Ulrich H¨
ohle
Lawrence N. Stout
Erich Peter Klement
3

4

Program Committee
Ulrich H¨
ohle, Wuppertal, Germany (Chairman)
Lawrence N. Stout, Bloomington, IL, USA (Chairman)
Ulrich Bodenhofer, Linz, Austria
Bernard De Baets, Gent, Belgium
Didier Dubois, Toulouse, France
J´anos Fodor, Budapest, Hungary
Llu´ıs Godo, Barcelona, Spain
Siegfried Gottwald, Leipzig, Germany
Petr H´ajek, Praha, Czech Republic
Erich Peter Klement, Linz, Austria
Wesley Kotz´e, Grahamstown, South Africa
Radko Mesiar, Bratislava, Slovakia
Daniele Mundici, Firenze, Italy
Endre Pap, Novi Sad, Serbia
Stephen E. Rodabaugh, Youngstown, OH, USA
Susanne Saminger-Platz, Linz, Austria
Aldo Ventre, Napoli, Italy
Siegfried Weber, Mainz, Germany
Local Organizing Committee
Erich Peter Klement, Johannes Kepler University Linz (Chairman)
Ulrich Bodenhofer, Johannes Kepler University Linz
Andrea Breslmayr, Johannes Kepler University Linz
Sabine Lumpi, Fuzzy Logic Laboratorium Linz-Hagenberg
Bernhard Moser, Software Competence Center Hagenberg
Susanne Saminger-Platz, Johannes Kepler University Linz
Isabel Tober-Kastner, Software Competence Center Hagenberg
Thomas Vetterlein, Johannes Kepler University Linz
5

6

Contents
Michael Bukatin, Ralph Kopperman, Steve Matthews
On the nature of correspondence between partial metrics
and fuzzy equalities ....................................................9
Mustafa Demirci
A Stone-type adjunction for fixed-basis fuzzy topological spaces
in abstract categories and its applications ...............................13
Jeffrey T. Denniston, Austin Melton, Stephen E. Rodabaugh
Enriched topological systems and variable-basis enriched functors ........16
Patrik Eklund, M. A. Gal´an, Robert Helgesson, Jari Kortelainen
Ontology <Logic or Ontology =Logic? ...............................21
Nikolaos Galatos, Jos´e Gil-F´erez
Closure operators on modules over quantaloids:
applications to algebraic logic .........................................25
John Harding, Carol Walker, Elbert Walker
Categories of fuzzy sets and relations ...................................28
Hans Heymans
Sheaves on involutive quantales: Grothendieck quantales .................29
Ulrich H¨ohle
Topology based on premultiplicative quantaloids:
a common basis for many-valued and non-commutative topology ..........31
S´andor Jenei, Franco Montagna
Two new classification theorems on residuated monoids ..................34
Erich Peter Klement, Anna Koles´arov´a, Radko Mesiar, Andrea Stupˇnanov´a
Level dependent capacities and integrals ................................41
Hongliang Lai, Dexue Zhang
Globalization of Cauchy complete preordered sets
valued in a divisible quantale ..........................................44
7

Steve Matthews, Michael Bukatin, Ralph Kopperman
Discrete partial metric spaces .........................................46
Fu-Gui Shi, Bin Pang
Categories isomorphic to L-fuzzy closure system spaces ..................48
Alexander ˇ
Sostak
Variable range categories of approximate systems . . . . . . . . . . . . . . . . . . . . . . . 52
Milan Stehl´ık
Category Theory in Statistical Learning? ...............................56
Isar Stubbe
Quantaloid-enriched categories for multi-valued logic and other purposes . 58
Thomas Vetterlein
On the characterisation of regular left-continuous t-norms ................59
Carol Walker, Elbert Walker
Type-2 operations on finite chains ......................................63
Yueli Yue
Convergence and compactness in fuzzy metric spaces . . . . . . . . . . . . . . . . . . . . 67
Dexue Zhang
Quantale-valued preorders as enriched categories . . . . . . . . . . . . . . . . . . . . . . . 71
8

On the nature of correspondence between
partial metrics and fuzzy equalities
Michael Bukatin1, Ralph Kopperman2, and Steve Matthews3
1Nokia Corporation
Boston, Massachusetts, USA
bukatin@cs.brandeis.edu
2Department of Mathematics
City College, City University of New York, New York, USA
rdkcc@ccny.cuny.edu
3Department of Computer Science
University of Warwick, Coventry, UK
Steve.Matthews@warwick.ac.uk
1 Introduction
The correspondence between partial metrics and fuzzy equalities was discovered in
2006 [1]. It was immediately apparent that there was a duality between metric and
logical viewpoints, and so the question about the nature of correspondence between
partial metrics and fuzzy equalities arose.
Initially, the authors of [1] suggested that we should talk about equivalence between
partial metrics and fuzzy equalities up to the choice of dual notation. This suggestion
was based on the notion that the duality between metric and logical viewpoints belonged
to the metalevel and was a part of the mindset of the practitioners in the respective fields,
but did not affect the mathematical structures involved. We refer to this suggestion as
the equivalence approach.
The equivalence approach remains a legitimate way of viewing this correspondence.
In particular, while there is a varierty of possible choices of allowed spaces and mor-
phisms, in all cases studied so far there are (covariant)isomorphisms of the correspond-
ing categories of partial metric spaces and spaces equipped with fuzzy equalities. The
induced specialization orders on a partial metric space and the corresponding space
equipped with a fuzzy equality also coincide. So, in this sense there seems to be no
duality between partial metrics and fuzzy equalities themselves.
Later Mustafa Demirci suggested that the duality between metric and logical view-
points should nevertheless be brought into formalization of this correspondence by ex-
plictly requiring that logical values and distances were respresented by dual structures.
We refer to this suggestion as the duality approach.
It turns out that the duality approach to understanding this situation is preferrable.
It allows to formally express a larger chunk of existing informal mathematical practice,
and it allows to do so without explicitly considering the metalevel. Even more impor-
tantly, being closer to the respective intuitions of the practitioners in the related fields
the duality approach makes it easier to develop applications.
9

Another aspect of the duality between logical values and distances is that the mul-
tiplicative notation is used on the logical side and the additive notation is used on the
metric side. This suggests that it might be possible to bring some kind of exponentia-
tion into play as well, potentially resulting in a more complicated correspondence and,
perhaps, a genuine duality between partial metrics and fuzzy equalities. To the best of
our knowledge, this has not been done so far and should be considered an open prob-
lem. (It should be noted here that it is not uncommon to start with a metric d(x,y), to
express the degree of similarity of xand yas f(x,y) = e−d(x,y), and to call the resulting
f(x,y)a fuzzy metric with the appropriate transformation of the axioms of a metric.)
2 Definitions
We provide informal sketches of definitions of quantale-valued partial metrics [3] and
quantale-valued sets (sets equipped with quantale-valued fuzzy equalities) [2].
2.1 Quantale-valued Partial Metrics
The quantale Vis a complete lattice with an associative and commutative operation +,
distributed with respect to the arbitrary infima. The unit element is the bottom element
0. The right adjoint to the map b7→ a+bis defined as the map b7→ b˙
−a=V{c∈
V|a+c≥b}. Certain additional conditions are imposed.
Definition 1. A V-partialpseudometric space is a set X equipped with a map p :X×
X→V (partial pseudometric) subject to the axioms
•p(x,x)≤p(x,y)
•p(x,y) = p(y,x)
•p(x,z)≤p(x,y) + (p(y,z)˙
−p(y,y))
2.2 Quantale-valued Sets
The quantale Mis a complete lattice with an associative and commutative operation ∗,
distributed with respect to the arbitrary suprema. The unit element is the top element
1. The right adjoint to the map b7→ a∗bis defined as the map b7→ a⇒b=W{c∈
V|a∗c⊑b}. Certain additional conditions are imposed.
Definition 2. An M-valued set is a set X equipped with a map E :X×X→M (fuzzy
equality) subject to the axioms
•E(x,y)⊑E(x,x)
•E(x,y) = E(y,x)
•E(x,y)∗(E(y,y)⇒E(y,z)) ⊑E(x,z)
10

3 Equivalence approach
Whenever we have a quantale in the sense of section 2.1, we can equip it with a dual
order, ⊑=≥, and it becomes a quantale in the sense of section 2.2 (and vice versa in the
opposite direction).
Define ∗as +,a⇒bas b˙
−a, 1 as 0 (and vice versa in the opposite direction).
Then partial pseudometrics and fuzzy equalities coincide as sets of functions. This
justifies the equivalence approach.
4 Duality approach
However we found it convenient to press the duality approach as far as possible.
4.1 Partial Ultrametrics Valued in Browerian algebras
For example, consider Ω-sets valued in Heyting algebras. Following the duality ap-
proach, on the metric side of things we will talk about partial ultrametrics valued in
dual Heyting algebras, but really pressing this approach as far as possible, we’ll use the
terminology ”partial ultrametrics valued in Browerian algebras”, and when Ωis actually
the algebra of open sets of a topological space X, we will consider partial ultrametrics
valued in the algebra of closed sets of the same space.
This helps to understand and establish the following result.
4.2 Sheaves of Sets as Co-sheaves of α-ultrametrics and Non-expansive Maps
Consider a complete Heyting algebra Ω. Consider a corresponding complete Browerian
algebra α.
Then every separated pre-sheaf of sets over Ωcan be thought of as a separated
co-pre-sheaf of α-ultrametrics and non-expansive maps over α.
To develop the necessary intuition one should first consider the case when Ωand α
are the algebras of, respectively, open and closed sets of a given topological space.
4.3 Partial Metrics into Non-negative Reals
In the logical situations (arising in domains for denotational semantics, and, in general,
in connection with the specialization order on the space of distances) we typically have
to flip the ray of non-negative reals, making 0 the top element.
If we press the duality approach as far as possible, the logical counter-part of the
partial metrics into non-negative reals ought to be fuzzy equalities valued in non-
positive reals. So instead of flipping the ray of non-negativereals we replace it with the
symmetric ray of non-positive reals.
Partial ultrametrics correspond to idempotent logic (usually, to the ordinary intu-
itionistic logic). Partial metrics should typically correspond to linear logic, and we think
about linear logic as the resource-sensitive logic. So, from the linear logic point of view,
it is natural to think about the weight (self-distance) of an element as the work which
still needs to be done to make it fully defined. This is the work to be done, something
owed, hence negative.
11

4.4 Intuition Related to Relaxed Metrics
Relaxed metrics typically map (x,y)into an interval number [l(x,y),u(x,y)], where uis
usually a partial metric, and lis usually a symmetric function, such that l(x,y)≤u(x,y).
Function uyields an upper bound for the inequality between “true, underlying x
and y”; essentially, “xand ydiffer no more that u(x,y)”, while lyields a lower bound
for that, essentially, “xand ydiffer at least by l(x,y)”. There is an intimate relationship
between land negative information, and also between land tolerances.
From the earlier logical considerations of relaxed metrics we know that udualizes,
but ldoes not. This means that on the logical side, Ubecames negative (non-positive,
actually), but Lremains non-negative.
So, while Urepresents a work still owed (a work to estimate distance better, actu-
ally), and hence negative, Lrepresents a work done, and hence positive (on the logical
side). Interestingly enough, the condition l(x,y)≤u(x,y)on the metric side becomes
L(x,y) + U(x,y)≤0 on the logical side.
If the distance between elements, xand y, is precesely defined (often the case for
maximal elements xand y), then l(x,y) = u(x,y), or equivalently L(x,y) + U(x,y) = 0,
expressing the fact that no further computations are owed.
In general the amount which expresses debt here is not U(x,y), but L(x,y)+U(x,y) =
l(x,y)−u(x,y). (Note that l(x,x)is always 0, so the self-distance is always fully owed.)
5 Conclusion
The correspondence between partial metrics and fuzzy equalities allows for the transfer
of results and methods between these field, and helps in considering non-trivial inter-
play between metric and logical situations.
There is a long list of situations where this correspondence should be useful. We
only name a few of them here.
It is particularly important to study metric counterparts of the logical research gener-
alizing the fuzzy equalities to the non-commutative case and to categories, in particular
results for sets valued in non-commutative quantales (H¨ohle and Kubiak) and results
for sets valued in Grothendieck topologies (Higgs).
Weighted quasi-metrics are a remarkably effective instrument on the metric side,
and their logical counterparts would probably be as useful as the global quantale-valued
sets which are the logical counterparts of weighted metrics.
References
1. Bukatin, M., Kopperman R., Matthews S., Pajoohesh H.: Partial Metrics and Quantale-valued
Sets. Preprint (2006); http://pages.cs.brandeis.edu/∼bukatin/distances and equalities.html.
2. H¨ohle U., M-valued sets and sheaves over integral, commutative cl-monoids. In: Rodabaugh,
S.E., et al (eds.) Applications of Category Theory to Fuzzy Subsets, pp. 33–72. Kluwer Aca-
demic Publishers, Dordrecht, Boston, London (1992).
3. Kopperman, R., Matthews, S., Pajoohesh, H.: Partial metrizability in value quantales. Applied
General Topology, 5(1), 115–127 (2004).
12

A Stone-type adjunction for fixed-basis fuzzy topological
spaces in abstract categories and its applications
Mustafa Demirci
Department of Mathematics
Akdeniz University, Antalya, Turkey
demirci@akdeniz.edu.tr
Abstract. In this study, fixed-basis fuzzy topological spaces are formulated on
the basis of a certain object of an abstract category, and a Stone-type adjunction
for them is established. Applications and consequences of this adjunction is dis-
cussed. As its particular consequence, it is shown that the category of B-categories
(and so the category of quantale preordered sets) is dually adjoint to the category
of base spaces.
1 Introduction and motivation
Since the inception of the fuzzy topological spaces (called the lattice-valued topological
spaces or the many valued topological spaces in more recent terminology), their truth
value structures (or their bases in the terminology of [8, 5, 12, 13]) have been extensively
studied in the literature. The selection of bases varies from author to author and from pa-
per to paper. Completely distributive complete lattices with order-reversing involutions
[16], semiframes [11], frames [6], cl∞-monoids [10], GL-monoids with square roots
[7], complete groupoids [5], complete quasi-monoidal lattices [8, 12], semi-quantales
[2, 13] and algebras in varieties [14] are known examples of such bases. The diversity
of bases naturally brings the question of how the notion of fixed-basis fuzzy topological
space can be defined on the basis of an object Lof an abstract category C. Although
a similar categorical question is also valid for other approaches to the notion of fuzzy
topological spaces such as variable-basis fuzzy topological spaces [11–13] and general-
ized lattice-valued topological spaces [2], we focus on only the fixed-basis case in this
study. Apart from fuzzy topologies, Stone-type adjunctions form an important theme
of the order-theory (see [3] and the references therein). Among others, the adjunction
between the category Loc of locales and the category Top of topological spaces is a
well-known example of these adjunctions. The studies on Stone-type adjunctions give
rise to a fundamental question: Is it possible to extend the adjunction between Loc and
Top to an adjunction between an abstract category Cand a category of spaces in some
generalized sense? This question is tantamount to the formulation of Stone-type ad-
junctions for abstract categories. Its solution relies on the fixed-basis fuzzy topological
spaces asked in the former question. The main aim of this study is to introduce the
notion of C-M-L-space as a categorical generalization of fixed-basis fuzzy topological
space being an answer to the former question, and is to construct a dual adjunction be-
tween Cand the category of C-M-L-spaces providing an answer to the latter question.
13

2 C-M-L-spaces and their dual adjunction with C
Let the category Chave products, and let Mbe a class of monomorphisms in C. Fur-
thermore, let us fix a C-object L.
Definition 1. For a set X, we call the pair X,τm
→LXaC-M-L-space, and τm
→LX
aC-M-L-topology on X iff τm
→LXis an M-morphism.
Proposition 1. Each function f :X→Y determines a unique C-morphism f ←
L:LY→
LX(called backward C-L-power operator of f ) such that πx◦f←
L=πf(x)for all x ∈X .
Definition 2. Given C-M-L-spaces X,τm1
→LXand Y,νm2
→LY, a function f :X→
Y is C-M-L-continuous iff there exists a C-morphism rf:ν→τfilling out the following
commuting diagram:
LYf←
L
−→ LX
m2↑ ↑ m1
νrf
−→ τ
C-M-L-spaces and C-M-L-continuous maps constitute a category that we denote
by C-M-L-SPC. By supplying examples, it will be justified that C-M-L-SPC is a cat-
egorical unification of many familiar categories of fixed-basis fuzzy topological spaces.
As the central result of this study, we will establish a categorical generalization of the
adjunction between Loc and Top in the following manner:
Theorem 1. For E⊆Mor(C)and M⊆Mon (C), let Chave (E,M)-factorizations
and the unique (E,M)-diagonalization property in the sense of [1]. Then Cis dually
adjoint to C-M-L-SPC.
Referring to the unit and co-unit of the adjunction Co p ⊣C-M-L-SPC, we define
L-spatial C-objects and L-sober C-M-L-spaces, and then point out in this study that the
restriction of the adjunction in Theorem 1 to the full subcategory of Cof all L-spatial
objects and the full subcategory of C-M-L-SPC of all L-sober objects gives a dual
equivalence between these subcategories. The adjunction Cop ⊣C-M-L-SPC covers
many known and new dual adjunctions between various kinds of ordered-structures and
various kinds of generalized topological spaces. Because of practical purposes, we pay
a special attention to the explicit determination of C-M-L-SPC for a concrete category
C. In particular, it will be proven in this talk to be an application of Theorem 1 that the
category Cat(B) of B-categories [9, 15] (and so the category p-Q-Set of pre-Q-sets [9])
is dually adjoint to the category BS of base spaces [4].
References
1. J. Ad´amek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, John Wiley &
Sons, New York, 1990.
2. M. Demirci, Pointed semi-quantales and lattice-valued topological spaces, Fuzzy Sets Syst.
161 (2010) 1224-1241.
14

3. M. Demirci, (Z1,Z2)-complete partially ordered sets and their representations by Q-spaces
(submitted).
4. M. Ern´e, General Stone duality, Topology and its Applications 137 (2004) 125-158.
5. U. H¨ohle, Many valued topology and its applications, Kluwer Academic Publishers, Boston,
2001.
6. U. H¨ohle, Fuzzy sets and sheaves. Part II: Sheaf-theoretic foundations of fuzzy set theory
with applications to algebra and topology, Fuzzy Sets Syst. 158 (2007) 1175-1212.
7. U. H¨ohle and A. ˇ
Sostak, A general theory of fuzzy topological spaces, Fuzzy Sets Syst. 73
(1995) 131-149.
8. U. H¨ohle and A. ˇ
Sostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. H¨ohle
and S. E. Rodabaugh (eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure The-
ory, The hand books of fuzzy sets series, Vol.3, Kluwer Academic Publishers, Boston, Dor-
drecht, 1999, pp. 273-388.
9. U. H¨ohle and T. Kubiak, A non-commutative and non-idempotent theory of quantale sets,
Fuzzy Sets Syst. 166 (2011) 1-43.
10. J.A. Goguen, The fuzzy Thychonoff theorem, J. Math. Anal. Appl. 43 (1973) 734-742.
11. S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets Syst. 40 (2) (1991) 297-
345.
12. S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: U. H¨ohle and
S. E. Rodabaugh (eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory,
The handbooks of fuzzy sets series, Vol.3, Kluwer Academic Publishers, Boston, Dordrecht,
1999, pp. 273-388.
13. S. E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topo-
logical theories for lattice-valued mathematics, International Journal of Mathematics and
Mathematical Sciences, Volume 2007 (2007), doi:10.1155/2007/43645.
14. S. A. Solovyov, Sobriety and spatiality in varieties of algebras, Fuzzy Sets Syst. 159 (2008)
2567-2585.
15. I. Stubbe, Categorical structures enriched in a quantaloid: Categories, distributions and func-
tors, Theory Appl. Categ. 14 (2005) 1-45.
16. L. Ying-Ming and L. Mao-Kang, Fuzzy topology, World Scientific, Singapore, 1997.
15

Enriched topological systems and
variable-basis enriched functors
Jeffrey T. Denniston1, Austin Melton2, and Stephen E. Rodabaugh3
1Department of Mathematical Sciences
Kent State University, Kent, Ohio, USA
jdennist@kent.edu
2Departments of Computer Science and Mathematical Sciences
Kent State University, Kent, Ohio, USA
amelton@kent.edu
3College of Science, Technology, Engineering, Mathematics (STEM)
Youngstown State University, Youngstown, Ohio, USA
rodabaug@math.ysu.edu
This work is motivated in part by a question arising from programming: if bitstream
xcorresponds to bitstream yto some degree α, and if bitstream ysatisfies predicate a
to some degree β, then would it not be appropriate to model the possibility that bit-
stream xsatisfies predicate ato at least some degree related to both αand β? The
current multi-valued literature on topological and other systems, e.g., [1, 2,4,5, 9,10],
ultimately rooted from [11], does not address this question. Notions from enriched cat-
egories help us address this question and its consequences.
An enriched category C[7] over a monoidal category (M,⊗,I,α,λ,ρ)[8] is a class
of objects with the following data (C1, C2, C3) subject to the following axioms (D1,
D2, D3), where in the latter the last applied compositions are from Mand the other
compositions come from (C3):
C1 ∀a,b∈C,∃!C(a,b)∈M(existence of hom-objects).
C2 ∀a∈C,∃ida:I→C(a,a)(identities).
C3 ∀a,b,c∈C,∃!◦abc :C(b,c)⊗C(a,b)→C(a,c)(composition of hom-objects).
D1 ∀a,b,c,d∈C,
(◦abd)◦(◦bcd ⊗1) = (◦acd)◦(1⊗ ◦abc)◦α.
D2 ∀a,b∈C,
λ= (◦abb)◦(idb⊗1).
D3 ∀a,b∈C,
ρ= (◦aab)◦(1⊗ida).
It is also said that Cis an M-enriched category.
It is well-known that a meet semilattice L(a poset closed under finite meets), taken
as a preordered category, is a (strict) monoidal category in which ⊗is the binary meet, I
is the top element ⊤, and the associator αand the unitors λ,ρare all identities. Further,
it can be shown:
16

Proposition 1. A set X, replacing Cabove, is an L-enriched category if and only if
there is an equality relation E on X such that:
E1 E :X×X→L is a mapping (degrees of correspondence).
E2 ∀x∈X,E(x,x) = ⊤(total existence).
E3 ∀x,y,z∈X,E(x,y)∧E(y,z)≤E(x,z)(transitivity).
It should be noted that each (Ci) corresponds precisely to each (Ei).
The consequent of the proposition is taken as the definition of (X,E)as an L-
enriched set.
If (E2) were to be replaced by a symmetry condition (∀x,y∈X,E(x,y) = E(y,x)),
then the Fourman-Scott definition [3] of an L-valued set would result as cited by H¨ohle
in [6].
For L-enriched set (X,E),E(x,y)is interpreted as the degree to which bitstream x
corresponds to bitstream y.
Finally, with an eye to variable-basis notions later, (X,E,L)is called an enriched
set.
Example 1. Examples of enriched sets include the following:
1. Let Xbe a set and Lbe a meet semilattice Lwith |L| ≥ 2.Choose a∈L− {⊤} and
put E:X×X→Lby
E(x,y) = a,x6=y
⊤,x=y.
Then (X,E,L)is an enriched set.
2. Let (X,d)be an ultrametric space bounded by 1, and put E:X×X→Lby
E(x,y) = 1−d(x,y).
Then (X,E,L)is an enriched set.
Given M-enriched categories Cand D,then F:C→Dis an M-enriched functor
[7] if the following hold:
F1 ∀a∈C,∃!F(a)∈D.
F2 ∀a,b∈C,∃!Fab ∈M(C(a,b),D(F(a),F(b))) .
F3 ∀a∈C,Faa ◦ida=idF(a)(in M).
F4 ∀a,b,c∈C,in Mit is the case that
Fac ◦(◦abc) = ◦F(a)F(b)F(c)◦(Fbc ⊗Fab).
Proposition 2. Given L-enriched sets (X,E)and (Y,F), where L is a meet semilattice,
it is the case that f :(X,E)→(Y,F)is an L-enriched functor if and only f :X→Y is
a mapping such that ∀x,y∈X,
E(x,y)≤F(f(x),f(y)).
17

The variable-basis extension of L-enriched functors makes use of monoidal functors
as defined in [8]. Let M,Nbe monoidal categories and let Cbe an M-enriched cate-
gory and Dbe an N-enriched category. Then (F,Ψ):C→Dis a(n) (variable-basis)
enriched functor if the following hold:
V0 Ψop :M←Nis a monoidal functor as defined in [8].
V1 ∀a∈C,∃!F(a)∈D.
V2 ∀a,b∈C,∃!Fab ∈M(C(a,b),Ψo p [D(F(a),F(b))]).
V3 ∀a∈C,Faa ◦ida=Ψo p idF(a)(in M).
V4 ∀a,b,c∈C,in Mit is the case that
Fac ◦(◦abc) = Ψop ◦F(a)F(b)F(c)◦(Fbc ⊗Fab).
The backward direction of, and notation for, Ψop are both motivated by topological
systems and variable-basis topology and, in particular, enriched topological systems
taken up below.
Proposition 3. Given enriched sets (X,E,L)and (Y,F,M), where L,M are meet semi-
lattices, it is the case that (f,ψ):(X,E,L)→(Y,F,M)is an enriched functor if and
only f :X→Y is a mapping and ψo p :L←M is a meet-semilattice morphism such that
∀x,y∈X,
E(x,y)≤ψop [F(f(x),f(y))] .
The proposition justifies the following definition:
Definition 1. The category EnrSet comprises enriched sets (X,E,L)as objects and
enriched functors (f,ψ)as morphisms; and in this setting, the latter are called enriched
mappings. The full subcategory in which each L is a frame and each ψis a localic
morphism is denoted EnrSetFrm.
It is straightforward that EnrSet and EnrSetFrm are categories using the composi-
tions and identities of Set and SLat(∧),the latter denoting the category of (finite) meet
semilattices and (finite) meet preserving mappings.
Enriched topological systems, namely topological systems based upon enriched
sets, can now be defined.
Definition 2. EnrTopSys has ground category EnrSetFrm ×Loc and comprises the
following data satisfying the following axioms:
1. Objects: ((X,L,E),A,),called enriched topological systems.
(a) (X,L,E)is an enriched set, A is a locale (ground conditions).
(b) is an L-satisfaction relation on (X,A),i.e., satisfies both arbitrary Wand
finite ∧interchange laws (topological system consition).
(c) E and are compatible, i.e., ∀x,y∈X,∀a∈A,E(x,y)∧(y,a)≤(x,a)
(compatibility condition).
2. Morphisms: (f,ψ,ϕ):((X,L,E),A,)→((Y,M,F),B,),called enriched con-
tinuous functions.
18

(a) (f,ψ):(X,E,L)→(Y,F,M)is an enriched mapping, ϕ:A→B is a localic
morphism (ground conditions).
(b) ∀x∈X,∀b∈B,(x,ϕop (b)) ≤ψo p ((f(x),b)) (partial adjointness).
3. Composition and identities: those of the ground EnrSetFrm ×Loc.
Both enriched topological systems and enriched continuous functions are in plenti-
ful supply, with a number of example classes at hand, including the following example
class.
Example 2. Each enriched set (X,L,E)with La frame generates an enriched topologi-
cal system. Given (X,L,E),put
τ=u∈LX:∀x,y∈X,E(x,y)∧u(y)≤u(x).
1. ∀y∈X,Ey:X→Lby Ey(x) = E(x,y).It follows that {Ey:y∈X} ⊂ τ.It is im-
portant to note that the proof makes explicit use of the transitivity condition (E3)
above.
2. The collection τcontains all constant L-subsets of X.
3. It follows from the infinite distributive law of Lthat τis an L-topology on Xand
hence a stratified L-topology on X.
Since Lis a frame, τis a locale. Now put :X×τ→Lby
(x,u) = u(x).
It can be checked that satisfies the arbitrary Wand finite ∧interchange laws and that
Eand are compatible. Hence ((X,E,L),τ,)is an enriched topological system.
Returning to the definition of an enriched topological system, certain comments
should be made. First, the compatibility condition addresses the question posed at the
beginning of this abstract. Second, it should be noted that partial adjointness is a signif-
icant weakening of the adjointness condition of Vickers [11] and the associated systems
literature, but it should also be noted that the inequality retained above from Vickers’
adjointness is a natural and important one from the standpoint of programming. These
considerations motivate weakening the adjointness condition for the morphisms of the
important category Loc-TopSys [2, 9,10] to partial adjointness as formally stated in the
above definition, thereby forming the category Loc-TopSys(≤).
Theorem 1. EnrTopSys maps functorially into Loc-TopSys (≤).
This theorem (with its proof) indicates that with respect to objects, traditional topo-
logical systems in the sense of Loc-TopSys already accommodate enriched topologi-
cal systems; but with respect to morphisms, Loc-TopSys must be generalized to Loc-
TopSys(≤)to accommodate enriched continuous functions between enriched topolog-
ical systems.
Finally, enriched topological systems afford new links to lattice-valued topology
and L-topological spaces in particular. For example, let ((X,L,E),A,)be an enriched
19

topological system. In addition to the already known frame map extL:A→LXand the
attendant L-topological space (X,ext→
L(A)),there is the frame map
ext(E,L):A→LX×Xby ext(E,L)(a) (x,y) = E(x,y)∧(y,a)
as well as, for fixed y∈Y, the frame map
ext(E,L,y):A→LXby ext(E,L,y)(a) (x) = E(x,y)∧(y,a).
Theorem 2. Let ((X,L,E),A,)be an enriched topological system. The following
hold:
1. ∀y∈Y,ext→
(E,L,y)(A)≺ext→
L(A), i.e., the former L-topology is a refinement of the
latter L-topology with respect to the ordering of LX
.
2. Within L(LX)
,it is the case that
ext→
L(A)⊂_
y∈X
ext→
(E,L,y)(A)≡**[
y∈X
ext→
(E,L,y)(A)++.
3. (X,ext→
L(A)) L-homeomorphically embeds into X×X,ext→
(E,L)(A),namely the
former is L-homeomorphic to the subspace ∆(X×X),hext→
(E,L)(A)i|∆(X×X).
References
1. J. T. Denniston, S. E. Rodabaugh, Functorial relationships between lattice-valued topology
and topological systems, Quaestiones Mathematicae 32:2(2009), 139–186.
2. J. T. Denniston, A. Melton, S. E. Rodabaugh, Interweaving algebra and topology: Lattice-
valued topological systems, Fuzzy Sets and Systems, to appear.
3. M. Fourman, D. S. Scott, Sheaves and logic, Applications of Sheaves: Lecture Notes in
Mathematics 753(1979), 302–401, Springer-Verlag (Berlin, Heidelberg, New York).
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E. P. Klement, Abstracts of the 30th Linz Seminar, Universit¨atsdirektion Johannes Kepler
Universit¨at, Linz, Austria, 3–7 February 2009, pp. 52–54.
5. C. Guido, Fuzzy points and attachment, Fuzzy Sets and Systems 161:16(2010), 2150–2165.
6. U. H¨ohle, Presheaves over GL-monoids, in U. H¨ohle, E. P. Klement, Non-Classical Logics
and their Applications to Fuzzy Subsets: Theory and Decision Library: Series B: Mathe-
matical and Statistical Methods 32(1995), Kluwer Academic Publishers (Boston, Dordrecht,
London), pp. 127–158.
7. G. M. Kelly, Basic Concepts of Enriched Category Theory, Reprints in Theory and Applica-
tions of Categories 10(2005).
8. S. Mac Lane, Categories for the Working Mathematician, second edition, Graduate Texts in
Mathematics 5(1998), Springer Verlag (Berlin, Heidelberg, New York).
9. S. A. Solovyov, Variable-basis topological systems versus variable-basis topological spaces,
Soft Computing 14:10(2010), 1059–1068.
10. S. A. Solovyov, Categorical foundations of variety-based topology and topological systems,
Fuzzy Sets and Systems, to appear.
11. S. J. Vickers, Topology Via Logic, Cambridge University Press (Cambridge, 1989).
20

Ontology <Logic or Ontology =Logic ?
Patrik Eklund1, M.A. Gal´an2, Robert Helgesson1, and Jari Kortelainen3
1Ume˚a University
Ume˚a, Sweden
{peklund, rah}@cs.umu.se
2University of M´alaga
M´alaga, Spain
3Mikkeli University of Applied Sciences
Mikkeli, Finland
jari.kortelainen@mamk.fi
An equivalent formulation of the title question is “Are terms important?” Yet an-
other isomorphic formulation is “Is concept (of) a type?”
Quantales are nice. Rosenthal [19] defined a (unital) quantale (Q,•,1,∨)to be a
monoid (Q,•,1)and a complete semilattice (Q,∨), so that •distributes over ∨. This
abstract was partly inspired by the LINZ2012 call for papers text “quantales and its ap-
plications to theoretical computer science”, yet, this is not an abstract about “quantales
and its applications to theoretical computer science”. What is this abstract then about?
It is about logic, it is about fuzzy and uncertainty representation, but in particular it is
also, but not only, and very much in particular not only, about truth values.
We are bold enough to say, that this abstract is not given justice, until the reader is
eventually at a saturated understanding about the main claim of this abstract (not saying
the reader has to agree with the authors on the claim), in fact being one important
main claims of our work during the past decade, ever since the underlying ideas behind
compositions involving the term monad [4] was presented at LINZ2000.
This main claim is stated in the following
Theorem 1. Yes, terms are important!
Initially we want to say something informally also about the other questions. It will be
clear that
Proposition 1. “Ontology <Logic” iff “Concept is a type”, and “Ontology = Logic”
iff “Concept is of a type”.
Corollary 1. “Being of a type” and “Being a type” is mutually exclusive.
1 Logic and fuzzy logic
Logic is not only computing with truth values. For propositional calculus, yes, but as
soon as we involve sentences with content as provided by terms, in turn building upon
an underlying signature, logic computation involves much more than mere manipulation
of truth values.
21

Fuzzy logic is in a simple view extensions of whatever is crisp. Traditionally, fuzzy
logic is extending crisp truth values to fuzzy truth values. Most of the fuzzy logic litera-
ture indeed does not go beyond fuzzification of anything else but truth values. Moreover,
approach like Hajek’s BL [14] do go on into predicates, but terms inside predicates are
left as crisp objects so that e.g. substitution still is the very traditional and crisp one.
The situation ‘Ontology <Logic’ appears typically in description logic, which as-
sumes concepts to be atomic, i.e. description logic appears more like a propositional
calculus than a predicate calculus. In fact, the underlying assumption seems very much
like having one single type concept, and having often a huge number of atomic con-
cepts, like e.g. seen in the medical vocabulary SNOMED CT, that like OWL/RDF has
adopted EL++ as a variant of description logic for its ontological purposes. The sim-
plicity of description logic is certainly intentional, as the motivation of using such a
‘partial logic’ is given the need to capture vocabulary, terminology and thesauri more
than explicitely reasoning with these concepts and structures.
However, were we to become interested in fuzzy ontology there is a risk that fuzzy
ontology in this narrow sense takes routes that even moves away from logical thinking.
Such fuzzy ontologies may later appear in fuzzy reasoning, and then it is not clear that
fuzzy approaches in fuzzyfying ontologies correlates with fuzzification of the logical
machineries.
This calls for using terms, and indeed assigning an important role to terms and their
semantics. Clearly, we also strongly speak in favour of terms in the wider sense, in
particular concerning uncertainty modelling of terms and and not just involving terms.
2 Terms in the wider sense
Terms are not interesting as such. Terms are interesting as part of sentences, and not to
forget, terms are interesting as part of other terms, the latter interest obviously leading
to substitution.
Terms are defined by a corresponding term monad, means that substitutions are
morphisms in the Kleisli category related to that particular term monad.
In [5, 6] we pointed a number of paradigms capturing different ways of modelling
uncertainty in these respects. These paradigms make a clear distinction between ‘op-
erating with fuzzy’ and ‘fuzzy operation’. The underlying term monad for the former
is the composition of the fuzzy powerset monad with the traditional term monad, and
doing all this over Set. The underlying term monad for the latter builds upon an endo-
functor over Set(Q), where in principle Qcould be a quantale, or could be something
else, yet appropriate. This gives us the basis for the “fully fuzzy” situation which has
it’s starting point in considerations for terms and substituting with terms. Note that truth
values of sentences have not yet entered the scene at all. Notably, one might even allow
oneself to have a crisp logic with “fully fuzzy” terms. In fact, in real life applications,
this is indeed what happens mostly, i.e. observations and assessments of data and infor-
mation are fuzzy, but decision-making, like in health care for interventions, must in the
end be crisp.
22

It should also be remarked that a shift from one-sorted to many-sorted is far from
trivial, even if folklore literature claims otherwise. Algebraic considerations need also
be precisely handled, as pointed out in [6].
Such terms then as included in sentences provides leads again to question and no-
tions about fuzzy sentences, and so on and so forth. The entire logic machinery all
the way down to inference calculi can be nicely described e.g. in the framework of
Meseguer’s general logic [16]. Moreover, general logic can be further generalized from
the viewpoint of Theorem 1, namely, that a substitution oriented generalized general
logic indeed is more than feasible, not to say very desirable [7, 8].
3 Type theory
Whereas for terms, informal definitions of the term set mostly correspond to the formal
definitions of terms, so that ambiguities are avoided. Concerning λ-terms, the situation
with informal definitions about what is and isn’t λ-terms is less obvious, in particular
in the typed case. In [8] we make this situation explicit by considering levels of signa-
tures, i.e. being very observant about where particular types and related operators reside
especially before and after λ-abstraction. Type constructors also need to be handled for-
mally, and their respective algebras must be identified with utmost care.
In this abstract we will not provide detail. However, we may say that starting from
a usual signature Σ= (S,Ω), identifying the underlying primitive operations, we have
the term monad TΣ, over Set, or fully fuzzy over Set(Q). This situation is signatures,
terms and algebras at level one.
Then we may create a new signature SΣ= ({type},Ω′), on signature level two,
with type as the only sort, and operators in Ω′to be understood as type constructors.
Interesting on level two is the algebra of type, namely, A(type)is the underlying
category of your choice.
Now we can make TSΣ∅the sort set for signature level three, and the interesting
part is defining some operators into this signature.
In this separation of levels it is very transparent how e.g. operators at level one are
shifted over to level three. The most important observation at this stage is that λis not
a ‘term transformer’ but an ‘operator mover’ between level one and level three.
All this notions can be made precise, and we are able to show e.g. how problems
with variable renaming can be avoided. This is fully developed in [8].
Acknowledgements. This abstract and all our work on invoking uncertainty modeling
using suitable underlying categories is truly inspired by Lawrence Neff Stout. Thanks,
Larry!
References
1. J. Ad´amek, H. Herrlich, G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats,
John Wiley & Sons, Inc., 1990.
2. J. Beck, Distributive laws, Seminars on Triples and Categorical Homology Theory, 1966/67,
Lecture Notes in Mathematics 80, Springer-Verlag, 1969, pp. 119 – 140.
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3. P. Eklund, W. G¨ahler, Fuzzy filter functors and convergence. In: S. E. Rodabaugh, E. P.
Klement, U. H¨ohle (eds.) Applications of category theory to fuzzy subsets, Kluwer Academic,
1992, 109 – 136.
4. P. Eklund, M. A. Gal´an, M. Ojeda-Aciego, A. Valverde, Set functors and generalised
terms, Proc. IPMU 2000, 8th Information Processing and Management of Uncertainty in
Knowledge-Based Systems Conference, vol. III, 2000, 1595-1599.
5. P. Eklund, M. A. Gal´an, J. Kortelainen, L. N. Stout, Paradigms for non-classical substitu-
tions, Proc. 39th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2009),
May 21-23, 2009, Naha, Okinawa (Japan), 77-79.
6. P. Eklund, M.A. Gal´an, R. Helgesson, J. Kortelainen, Paradigms for many-sorted non-
classical substitutions, 2011 41st IEEE International Symposium on Multiple-Valued Logic
(ISMVL 2011), 318-321.
7. P. Eklund, R. Helgesson, Monadic extensions of institutions, Fuzzy Sets and Systems, 161
(2010), 2354-2368.
8. P. Eklund, R. Helgesson, Substitution logic as monadic extension of general logic, to be
submitted.
9. P. Eklund, M. A. Gal´an, J. Medina, M. Ojeda Aciego, A. Valverde, Powersets of terms and
composite monads, Fuzzy Sets and Systems, 158 (2007), 2552-2574.
10. P. Eklund, J. Kortelainen, L. N. Stout, Introducing fuzziness in monads: cases of the term
monad and the powerobject monad, Abstract in LINZ2009, 30th Linz Seminar on Fuzzy Set
Theory, Linz, Austria, February 3-7, 2009.
11. P. Eklund, J. Kortelainen, L. N. Stout, Adding fuzziness using a monadic approach to terms
and powerobjects, Fuzzy Sets and Systems, accepted.
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gences Structures, and Their Applications, Dubrovnik (Yugoslavia) 1987, Plenum Press,
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drecht, 2002, 312-314.
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19. K. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics
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S. E. Rodabaugh, E. P. Klement, U. H¨ohle (eds.), Applications of Category Theory to Fuzzy
Subsets, Kluwer Academic, 1992, pp. 74 – 106.
21. L. N. Stout, Categorical approaches to non-commutative fuzzy logic, Fuzzy Sets and Sys-
tems, 161 (2010), 2462-2478.
24

Closure operators on modules over quantaloids:
applications to algebraic logic
Nikolaos Galatos1and Jos´e Gil-F´erez2
1Department of Mathematics
University of Denver, Denver, Colorado, USA
ngalatos@du.edu
2Institute of Computer Science
Academy of Science of the Czech Republic, Prague, Czech Republic
josegilferez@gmail.com
Motivated by the study of the sentential logics and π-institutions, we introduce
the notion of closure operator on modules over quantaloids, and always driven by the
search of the solution of the Isomorphism Problem, as we explain below, we intro-
duce the notions of interpretability and representability between closure operators. This
yields a very rich theory with many nice properties: We prove that there exists a dual-
ity in the categories of modules over quantaloids, that they are strongly complete and
strongly cocomplete, that they are (Epi, Mono)-structured regular categories, that they
have enough injectives and projectives, and that they satisfy the strong amalgamation
property, among others. Some of these results are generalizations of the same results
obtained by Solovyov for categories of modules over quantales (see [10]). We charac-
terize monos and epis in the categories of modules over quantaloids, and furthermore
prove that every epi is induced by a closure operator on its domain.
We also study the notions of closure system on a module over a quantaloid, and
prove that they are exactly the submodules of the dual module, and that the standard
correspondence between closure operators and closure systems on a set extends to a
natural isomorphism. We prove that the set of closure operators that are interpretable
by a given morphism τis a principal filter of the lattice of closure operators on its
domain. As a consequence, we obtain that every extension of an interpretable closure
operator is also interpretable by the same morphism. One instantiation of this result is
the well-known fact (see Theorem 2.15 of [4]) that if a sentential logic has an algebraic
semantics, then every extension of it also has an algebraic semantics and with the same
defining equations.
The Problem of the Isomorphism has its origin in the work of Blok and J´onsson,
who in order to study the property of algebraizability for sentential logics, and the
equivalence between deductive systems in general, introduced the notion of equivalence
between structural closure operators on a set Xacted on by a monoid M, or an M-set
(see [1]). As usual, given a monoid (M,·,1), an M-set consists of a set Xand a monoid
action ⋆:M×X→X, where 1 ⋆x=xand a⋆(b⋆x) = (a·b)⋆x, for all a,b∈Mand
x∈X. While the use of closure operators to encode entailment relations is very well
known, the action of the monoid is introduced to formalize the notion of structurality,
25

that is, “entailments are preserved by uniform substitutions,” a property usually required
for logics.
Given an M-set hX,·i, a closure operator Con Xis structural on hX,·i if and only if
it satisfies the following property: for every σ∈M, and every Γ⊆X,σ·CΓ⊆C(σ·Γ),
where σ·Γ={σ·ϕ:ϕ∈Γ}. This can be shortly written as follows:
∀σ∈M,σC6Cσ.(Str)
This is known as the structurality property forC, since it takes the following form, when
expressed in terms of ⊢C, the closure relation on Xassociated with the closure operator
C(defined by ϕ∈CΓiff Γ⊢Cϕ): for every Γ⊆X, every ϕ∈X, and every σ∈M,
Γ⊢Cϕ⇒σ·Γ⊢Cσ·ϕ.
For every σ∈M, a unary operation Cσon Cl(C) = hCl(C),⊆i, the lattice of the-
ories or closed sets of C, is defined in the following way: Cσ(Γ) = C(σ·Γ). The ex-
panded lattice of theories of a structural closure operator Cis defined as the structure
hCl(C),(Cσ)σ∈Mi.
In their approximation, Blok and J´onsson define two structural closure operators on
two M-sets to be equivalent if their expanded lattices of theories are isomorphic. Later,
they prove that under certain hypotheses (the existence of basis), this is equivalent to
the existence of conservativeand mutually inverse interpretations, which is the original
idea of equivalence between deductive systems emerging from the work of Blok and
Pigozzi. This equivalence between the lattice-theoretic property of having isomorphic
expanded lattices of theories, and the semantic property of being mutually interpretable
is known by the name of the Isomorphism Theorem. And the problem of determining
in which situations there exists an Isomorphism Theorem is called the Isomorphism
Problem.
The first Isomorphism Theorem was proved by Blok and Pigozzi in [2] for alge-
braizable sentential logics, and later it was obtained for k-dimensional deductive sys-
tems by them in [3] and for Gentzen systems by Rebagliato and Verd in [9]. But there is
not a general Isomorphism Theorem for structural closure operators on M-sets, as there
are counterexamples for that (see [8]).
In turn, Voutsadakis studied in [11] the notion of equivalence of π-institutions at
different levels (quasi-equivalence and deductive equivalence) and identified term π-
institutions, for which a certain kind of Isomorphism Theorem also holds. The notion
of π-institution was introduced by Fiadeiro and Sernadas in their article [5] and can be
viewed as a generalization of deductive systems allowing multiple sorts. They constitute
a very wide categorical framework embracing sentential logics, Gentzen systems, etc.,
as they include structural closure operators on M-sets as a particular case. Therefore, a
general Isomorphism Theorem for π-institutions is not possible (see [7]).
Sufficient conditions for the existence of an Isomorphism Theorem were provided
in [8] and [7] for structural closure operators on M-sets (and graduated M-sets), and π-
institutions that encompass all the previous known cases. The first complete solution of
the Isomorphism Problem was found for closure operators on modules over residuated
complete lattices, or quantales (see [6]). In this article, the modules providing an Iso-
morphism Theorem are identified as the projective modules. In particular, cyclic projec-
tive modules are characterized in several ways, from which the Isomorphism Theorem
26

for k-deductive systems follows, and also for Gentzen systems, using that coproducts of
projectives are projective. The Isomorphism Problem for π-institutions remained open.
One of our main results, as an application of the theory of closure operators on
modules over quantaloids to Algebraic Logic, is the following:
Theorem 1. If Qis a quantaloid, then a Q-module P is projective if and only if every
representation of a closure operator on P into another closure operator is induced.
This is the key result to establish that every equivalence between two closure op-
erators on projective modules is induced by mutually inverse interpretations. That is
the general solution for the Isomorphism Problem in the setting of modules over quan-
taloids.
We also explain how every π-institution induces a closure operator on a module
over a quantaloid, and every translation between π-institutions induces a morphism in
the fibered category of all modules over quantaloids. Thus, we show how the theory
of closure operators on modules over quantaloids is a generalization of the theory of
interpretations and representations of π-institutions.
Acknowledgment. The second author was supported by grant P202/10/1826 of the
Czech Science Foundation.
References
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9. J. Rebagliato and V. Verd´u. Algebraizable Gentzen systems and the deduction theorem for
Gentzen systems. Mathematics Preprint Series 175, University of Barcelona, June 1995.
10. Sergey A. Solovyov. On the category Q-mod.Algebra Universalis, 58(1):35–58, 2008.
11. G. Voutsadakis. Categorical abstract algebraic logic: Equivalent institutions. Studia Logica,
Special Issue on Abstract Algebraic Logic, Part II, 74(5):275–311, 2003.
27

Categories of fuzzy sets and relations
John Harding, Carol Walker, and Elbert Walker
Department of Mathematical Sciences
New Mexico State University, Las Cruces, New Mexico, USA
{jharding, hardy, elbert}@nmsu.edu
We define a category whose objects are fuzzy sets and whose maps are relations
subject to certain natural conditions. We enrich this category with additional structure
coming from t-norms and negations on the unit interval. We develop the basic properties
of this category and consider its relation to other familiar categories.
References
1. S. Abramsky, B. Coecke, A categorical semantics of quantum protocols, Proceedings of the
19th Annual IEEE Symposium on Logic in Computer Science (LiCS’04), IEEE Comput. Soc.,
New York (2004) 415-425.
2. J. Harding, A link between quantum logic and categorical quantum mechanics, Int. J. Theor.
Phys 48 (2009) 769-802.
3. H. Herrlich, G. E. Strecker, Category Theory, Second Edition, Sigma Series in Pure Mathe-
matics, Heldermann Verlag Berlin (1979).
4. H. Nguyen, E. Walker, A First Course in Fuzzy Logic, Third Edition, CRC Press, Boca Raton,
FL (2006).
5. C. Walker, Categories of fuzzy sets, Soft Computing 8(4) (2003) 299-304.
28

Sheaves on involutive quantales:
Grothendieck quantales
Hans Heymans
Department of Mathematics and Computer Science
University of Antwerp, Antwerpen, Belgium
hans.heymans@ua.ac.be
1 Introduction
It is well known that sheaves on a topological space (X,O)give rise to a category called
a Grothendieck topos [7], which can be seen as a constructive universe of sets. The local
sections of a sheaf Fexist locally at opens of X, such that the subobjects of Fform a
complete Heyting algebra (or frame), not necessarily a Boolean algebra. This sketches
a rough idea of the link between logic and geometry, which is so fruitfully exploited in
topos theory.
Another important field is non-commutative geometry [2], in which geometry is
dealt with implicitly through the study of non-commutative algebras, like C∗-algebras.
Attempts to make the hidden non-commutative topology more explicit have led to sev-
eral formalisms, including the theory of (involutive) quantales [8, 11]. Frames, like the
lattice of opens of a topological space, are commutative idempotent quantales (with
a trivial involution). It is not a surprise that people started thinking about sheaves on
quantales.
This idea sounds very natural, but there is a certain risk involved: are quantales
really good candidates for non-commutative topology and can we find a definition of
sheaves on a quantale that encapsulates C∗-algebras? Unfortunately, this is still a matter
of discussion, after almost thirty years of research.
Although older definitions of sheaves on quantales (e.g., [9]) may diverge, more
recent versions are based on the observation that sheaves on a locale (frame) Ocan be
presented in the form idempotent symmetric matrices with values in O[3]. The indices
of the matrix represent the local sections and the values of the matrix give the regions
in which pairs of local sections agree. By replacing the frame by an involutive quantale
Q, we obtain Q-valued sets. Many more references can be found in the recent paper of
Resende [10].
2 Enrichment over involutive quantaloids
The matrix approach is elegant, but problems emerge when one tries to conceptualize
the sheafification of Q-valued sets. By considering Q-valued sets as enriched categories
[1], we obtain more insight in these matters. They resemble metric spaces, which can be
29

considered as categories enriched over the quantale of positive real numbers (extended
with infinity). Some caution is in order: Q-valued sets are not categories enriched over
Q, but rather over an involutive quantaloid QE, obtained by splitting a certain class E
of idempotents of Q. Alternatively phrased, Q-valued sets are rather reflexive, transitive
and symmetric matrices with values in QE(i.e., symmetric monads or equivalence re-
lations). Having settled this, the sheafification of Q-valued sets may be defined as the
Cauchy completion of QE-categories ([14] is an early example). Many elements of en-
riched category theory contribute to sheaf theory (distributors [12], limits, etc.). On the
other hand, sheaves on an involutive quantale Qcan be cast in the form of modules over
Q[13, 6,5]. The more lattice theoretic oriented module theory has several advantages.
3 Grothendieck quantales
The sheaves on a locale give a localic Grothendieck topos. What about non-localic
Grothendieck toposes? We will show that every Grothendieck topos can be seen as the
category (allegory [4]) of sheaves on what we call a Grothendieck quantale. A plausible
definition of a Grothendieck quantale might be: an involutive quantale such that the
category of sheaves on it is a topos (this definition is slightly simplified). The main
result of the talk is a simple axiomatization of Grothendieck quantales [5]. If there is
time left, I would like to address some of the questions raised in the introduction.
References
1. R. Betti, A. Carboni, R. H. Street and R. F. C. Walters. Variation through enrichment. Journal
of Pure and Applied Algebra, 29:109–127, 1983.
2. A. Connes. Noncommutative Geometry. Academic Press Inc., San Diego, CA, 1994.
3. M. P. Fourman and D. S. Scott. Sheaves and logic. Lecture Notes in Mathematics, 753:302–
401, 1979.
4. P. J. Freyd and A. Scedrov. Categories, Allegories. North-Holland Mathematical Library 39,
Amsterdam, 1990.
5. H. Heymans. Sheaves on Quantales as Generalized Metric Spaces. PhD thesis, University of
Antwerp, 2010.
6. H. Heymans and I. Stubbe. Modules on involutive quantales: canonical Hilbert structure,
applications to sheaf theory. Order, 26(2):177–196. (arXiv:0809.4336v2)
7. S. Mac Lane and I. Moerdijk. Sheaves in Geometry and Logic: a First Introduction to Topos
Theory. Springer-Verlag, 1992.
8. C. J. Mulvey. ‘&’. Rendiconti del Circolo Matematico di Palermo, 12:99–104, 1986.
9. M. Nawaz. Quantales, quantal sets. PhD thesis, University of Sussex, 1985.
10. P. Resende. Groupoid sheaves as quantale sheaves. Journal of Pure and Applied Algebra,
216:41–70, 2012. (arXiv:0807.4848v3)
11. K. I. Rosenthal. Quantales and Their Applications. Pitman Research Notes in Mathematics
Series 234, Longman, 1990.
12. I. Stubbe. Categorical structures enriched in a quantaloid: categories, distributors and func-
tors. Theory and Applications of Categories 14:1–45, 2005.
13. I. Stubbe. Q-modules are Q-suplattices. Theory and Applications of Categories, 19:50–60,
2007.
14. R. F. C. Walters. Sheaves on sites as Cauchy complete categories. Journal of Pure and Ap-
plied Algebra, 24:95–102, 1982.
30

Topology based on premultiplicative quantaloids:
a common basis for many-valued and
non-commutative topology
Ulrich H¨ohle
Department of Mathematics and Natural Sciences
Bergische Universit¨at
Wuppertal, Germany
uhoehle@uni-wuppertal.de
Abstract. The purpose of this talk is to explain that topological spaces can be
formulated in any framework of premultiplicative quantaloid. In particular, the
following results are obtained in a cooperation with Tomasz Kubiak (Pozna´n,
Poland) during summer 2011.
Let Qbe a quantaloid ([2]). First we recall the cocompletion of Q-enriched cate-
gories (so-called Q-categories (cf. [3])) and specify the power Q-category monad TP
which is hidden behind the concept of cocompletion. Then we have the following the-
orems.
Theorem 1. Let P(X)X
w
ξbe a Q-functor. Then the following assertions are
equivalent:
(i) (X,ξ)satisfies the first algebra axiom — i.e. ξ·ηX=1X.
(ii) (X,ξ)is a TP-algebra.
Theorem 2. Let Xbe a skeletal Q-category and P(X)be the power Q-category. Then
the following assertions are equivalent:
(i) Xis cocomplete.
(ii) There exists a Q-functor P(X)X
w
ξsatisfying the first algebra axiom w.r.t.
the power Q-category monad.
After these preparations we introduce the concept of premultiplicative quantaloids.
Definition 1. A quantaloid Qis called premultiplicative if every hom-set
Q(a,b)has an binary operation ⊙satisfying the following conditions:
(pm1) ⊙is distributive over non empty joins in both variables,
(pm2) ⊙is subdistributive over the composition in both variables — i.e. for all
a,b,c∈obj(Q)and α,β∈Q(a,b)the subsequent relations are valid:
γ·(α⊙β)≤(γ·α)⊙(γ·β),γ∈Q(b,c)
(α⊙β)·γ≤(α·γ)⊙(β·γ),γ∈Q(c,a).
31

In this context ⊙is a called a premultiplication.⊓⊔
Example 1. Let [0,1]be the real unit interval equipped with the usual ordering and with
Łukasiewicz’ arithmetic conjunction ∗— i.e.
α∗β=max(α+β−1,0),α,β∈[0,1].
Obviously, ([0,1],∗)is a unital quantale. Further, let Qbe the quantaloid with one ob-
ject determined by ([0,1],∗). Then Qis a premultiplicative quantaloid w.r.t. the binary
minimum as well as w.r.t. the binary arithmetic mean.⊓⊔
Example 2. Let (L,
′)be a complete De Morgan algebra — this means that Lis a com-
plete (not necessarily distributive) lattice provided with an order reversing involution ′.
In particular, the universal upper (resp. lower) bound in Lis denoted by ⊤(resp. ⊥).
Then we construct a quantaloid Qas follows. The set of objects of Qis given by L
enlarged by a further element ω— i.e.
obj(Q) = L∪ {ω}.
The hom-sets of Qwith their respective partial orderings are given by:
–Q(a,a)is the two-point lattice for all a∈L∪ {ω}.
–Q(a,b)is a singleton, if a,b∈Lwith a6=b.
–Q(ω,b) = {λ∈L|λ≤b}with the ordering from L, if ω6=b.
–Q(a,ω) = {λ∈L|a′≤λ}with the ordering from Lop , if a6=ω.
Then there exists a unique composition law satisfying the following properties:
– The composition preserves arbitrary joins in each variable separately.
– On Q(a,a)the composition is the meet of the two-point lattice.
– If a6=band b6=c, then the composition attaches the universal lower bound of
Q(a,c)to all (λ1,λ2)∈Q(a,b)×Q(b,c).
Finally, the multiplicative structure on Qis determined as follows: On Q(a,a)we
use again the meet, while on hom-sets consisting of a unique morphism the binary
operation is evident. In order to complete the situation we have only to define binary
operations on Q(ω,b)and Q(a,ω)with a,b∈L:
λ1⊙b
ωλ2=(λ1,λ26=⊥,
⊥,λ2=⊥.
λ1⊙ω
aλ2=(λ2,λ16=⊤,
⊤,λ2=⊤.
All this shows that Qis a premultiplicative quantaloid. ⊓⊔
Let Qbe a premultiplicative quantaloid with the local premultiplication ⊙and
Cat(Qbe the category of Q-categories and Q-functors. We fix a Q-category X=
(X,e,d)and consider the Q-functor ⊡:P(X)×P(X)→P(X)induced by ⊙. Further,
let 1P(X)
w
⊤Xbe a Q-functor defined by:
⊤X(a) = (a,f⊤
a),f⊤
a(x) = _Q(a,e(x)),x∈X,a∈obj(Q).
32

An extremal subobject UP(X)
w
ιof P(X)is called a topology on Xiff ι
satisfies the following axioms:
(T1) ⊤Xfactors through ι.
(T2) ⊡·(ι×ι)factors through ι.
(T3) µX·P(ι)factors through ι.
The axiom (T1) means that ‘the whole space is open. (T2) is the intersection axiom and
(T3) means that ιis closed under internal joins — i.e. ιis cocontinuous.
If Xis provided with a topology ι, then (X,ι)is called a topological space in the
sense of the quantaloid Q.
Topological spaces in the sense of Qform a category which is topological over
Cat(Q).
In the case of Example 1 topological spaces are many valued topological spaces (cf.
[1]), while in the case of Example 2 we obtain non-commutative topological spaces pro-
vided the underlying De Morgan algebra is given by the lattice of all closed subspaces
of an arbitrary Hilbert space.
References
1. U. H¨ohle and T. Kubiak, Many valued topology and lower semicontinuity, Semigroup Forum
75 (2007), 1–17.
2. K. I. Rosenthal, The Theory of Quantaloids, Pitman Research Notes in Mathematics, vol.
348, Longman Scientific & Technical, Longman House, Burnt Mill, Harlow, 1996.
3. I. Stubbe, Categorical structures enriched in a quantaloid: categories, distributors and func-
tors, Theory and Applications of Categories 14 (1) 2005; pp. 1–45.
33

Two new classification theorems on residuated monoids
S´andor Jenei1,2and Franco Montagna3
1Institute of Mathematics and Informatics
University of P´ecs, P´ecs, Hungary
jenei@ttk.pte.hu
2Department of Knowledge-Based Mathematical Systems
Johannes Kepler University, Linz, Austria
sandor.jenei@jku.at
3Department of Mathematics and Computer Sciences
University of Siena, Italy
montagna@unisi.it
Abstract. We present two very recent Mostert-Shields style classification the-
orems on residuated l-monoids along with some related results in substructural
logics.
1 Introduction
Residuated lattices have been introduced in the 30s of the last century by Ward and
Dilworth [30] to investigate ideal theory of commutative rings with unit. Examples of
residuated lattices include Boolean algebras, Heyting algebras [6], MV-algebras [3],
basic logic algebras, [8] and lattice-ordered groups; a variety of other algebraic struc-
tures can be rendered as residuated lattices. The topic did not become a leading trend
on its own right back then. Nowadays the investigation of residuated lattices (that is,
residuated monoids on lattices) has got a new impetus and has been staying in the fo-
cus of strong international attention. The reason is that residuated lattices turned out to
be algebraic counterparts of substructural logics [27,26]. Applications of substructural
logics and residuated lattices span across proof theory, algebra, and computer science.
An extensive monograph discussing residuated lattices went to print in 2007 [7]. Sub-
structural logics encompass among many others, classical logic, intuitionistic logic,
relevance logics, many-valued logics, t-norm-based logics, linear logic and their non-
commutative versions. These logics had different motivations, different methodology,
and have mainly been investigated by isolated groups of researchers. The theory of
substructural logics has put all these logics, along with many others, under the same
motivational and methodological umbrella. Residuated lattices themselves have been
the key component in this remarkable unification.
Residuated lattices on the real unit interval [0,1]are of particular interest. On [0,1],
FLe-monoids (see Definition 1) are referred to as uninorms, integral FLe-monoids are
referred to as t-norms. Because they are residuated, those uninorms and t-norms ∗
are left-continuous, as two-place functions. The residuum →is given by x→y=
sup{z:z∗x≤y}. They determine both a substructural logic (obtained by interpreting
conjunction as ∗and implication as →) and a variety of commutative, integral and
34

bounded residuated lattices, see [7]. It follows that left-continuous uninorms originate
a substructural logic, which may lack not only contraction, but also weakening.
Both for left-continuous t-norms and for left-continuous uninorms, those with an
involutive negation are of special interest. (Note that for t-norms negation is defined
by ¬x=x→0, while for uninorms negation is defined as ¬x=x→f, where fis
a fixed, but arbitrary element of [0,1], and stands for falsum just like 0 does in case
of a t-norm). Involutive t-norms and uninorms have very interesting symmetry proper-
ties [11, 14,10, 24] and, as a consequence, for involutive t-norms and uninorms we have
beautiful geometric constructions which are lacking for general left-continuous t-norms
and uninorms [12, 20,23]. Furthermore, not only involutive t-norms and uninorms have
very interesting symmetry properties, but their logical calculi have important symmetry
properties too: Both sides of a sequent may contain more than one formula, while (hy-
per)sequent calculi for their non-involutivecounterparts admit at most one formula on
the right.
A particularly interesting question is whether the variety of algebras of a certain
logic are generated by only the algebras on [0,1]which are called standard algebras. If
the answer is yes, we say that the logic in question admits standard completeness. For
the logics BL and MTL this problem has been solved in [2] and [17], respectively.
As for the classification problem of residuated lattices, this task seems to be possible
only by posing additional conditions. The first result in this direction is due to Mostert
and Shields who investigated certain topological semigroups on compact manifolds
with connected, regular boundary in [28]. Being topological means that the monoid
operation of the residuated lattice is continuous with respect to the underlying topol-
ogy. They proved that such semigroups are ordinal sums in the sense of Clifford [4] of
product, Boolean, and Łukasieticz summands.
Next, the dropping of the topologically connected property of the underlying chain
can successfully be compensated by assuming the divisibility condition (which is, in
fact, the dual notion of the well-known naturally ordered property). The divisibility
condition is the algebraic analogue of the Intermediate Value Theorem in real analysis,
and it can be considered a stronger version of continuity of the monoidal operation:
Indeed, on a finite chain the order topology is the discrete one, so every operation is
continuous and hence does not necessarily obey the divisibility condition. Under the
assumption of divisibility, residuated chains, that is BL-chains, have been classified,
again, as ordinal sums with product, Boolean, and Łukasieticz summands. The divisi-
bility condition proved to be strong enough for the classification of residuated lattices
over arbitrary lattices too [22]. Fodor has classified those uninorms which have continu-
ous underlying t-norm and t-conorm [5]. But divisibility aside, no classification seemed
to be likely to exist due to the richness of residuated structures.
In this paper a first step is made in this direction: In one of the two classification
theorems of ours we do not assume divisibility nor even the slightest version of conti-
nuity.
First of all, we classify strongly involutive uninorms algebras (SIU-algebras), that is
bounded, representable, sharp, involutive FLe-monoids over arbitrary lattices for which
their cone operations are dually isomorphic. Let us remark that assuming the duality
35

condition proved to be equivalent to assuming the divisibility condition only for the
positive and negative cones of such algebras.
Second, we classify sharp involutive FLe-monoids on complete, order-dense, semi-
separable chains. Here neither divisibility nor even the weakest form of continuity is
assumed. Surprisingly, the restriction of those monoids to their negative cone is neces-
sarily continuous with respect to the order topology of their underlying chain. The result
seems only to hold under the condition t=f, and hence a classification for involutive
FLe-monoids is still lacking, but in any case the result is very surprising, as involutive
integral monoids may have discontinuities even below the fixed point of their negation.
While for involutive integral monoids (and even for involutive t-norms) a classifica-
tion is still lacking, for sharp involutive FLe-monoids on complete, order-dense, semi-
separable chains we can present a classification. Since [0,1]is a complete, order-dense,
semi-separable chain, our result provides with the classification of sharp, involutive
uninorms too. Remarkably, the adding of the involution condition to residuated integral
monoids does not bring us any closer to the solution of the related classification prob-
lem: As revealed by the rotation construction [12], every residuated integral monoid can
arise as a subsemigroup of an involutive residuated integral monoid.
Third, from the logical point of view, we want to solve some standard completeness
problems. Since uninorm logics are algebraizable in the sense of Blok and Pigozzi [1],
we can state the standard completeness problem in an algebraic way,recalling that valid
equations correspond to theorems of the associated logic and valid quasiequations cor-
respond to provable consequence relations. Now the question is if there is an equation
(resp., a quasiequation) of sharp, involutive, representable4FLe-monoids which is valid
in all sharp, involutive FLe-monoids on [0,1]but not in all sharp, representable invo-
lutive FLe-monoids? When such an equation (resp., quasiequation) does not exist, the
corresponding logic is standard complete (resp., finitely strongly standard complete).
In [25], it is shown that the logic of uninorm algebras is standard complete, and the
problem has been left open for the logic of involutive uninorm algebras (aka. bounded,
representable, sharp, involutive FLe-algebras). We prove that the logic of sharp, involu-
tive uninorm algebras is not standard complete and that the logic of involutive uninorm
algebras is not finitely strongly standard complete. In addition, we axiomatize the logic
of SIU-algebras and prove that it is finitely strongly complete with respect to the class
of standard SIU-algebras, it is not strongly complete with respect to the class of all
standard SIU-algebras, and that tautologicity and consequence relation in it are co-NP
complete.
2 Preliminaries
As said in the introduction, uninorms are commutative, isotone monoids on [0,1]. On
general universe, however, we shall refer to them as FLe-monoids:
Definition 1. Call U=hX,∗◦,≤,t,fiand as well its monoidal operation ∗◦ an FLe-
monoid if C=hX,≤i is a poset and ∗◦ is a commutative, residuated monoid over C
with neutral element t. Define the positive and the negative cone of Uby X+={x∈
4An FLe-monoid is representable if it is subdirect product of chains.
36

X|x≥t}and X−={x∈X|x≤t},respectively. Call an FLe-monoid Uinvolu-
tive, if for x∈X,(x′)′=xholds, where x′=x→∗◦ f. Call an involutive FLe-monoid
Usharp, if t=f. Call a sharp, involutive FLe-monoid a SIU-algebra, if for x,y∈X−,
x′∗◦ y′= (x∗◦ y)′holds.
Standing notation: For an FLe-monoid hX,∗◦,≤,t,fi, throughout the paper we de-
note the negative and the positive cone operation of ∗◦, by ⊗and ⊕, respectively.
Let Ube an FLe-monoid. The algebra U, and as well ∗◦ is called conic if every
element of Xis comparable with t, that is, if X=X+∪X−.Uis called finite if Xis a
finite set, Uis called bounded if Xhas top ⊤and bottom ⊥element. If Xis linearly
ordered, we speak about FLe-chains. Since ∗◦ is residuated, it is as well partially-ordered
(isotone), and therefore, ′:X→Xis an order-reversing involution. A partially-ordered
monoid is called integral (resp. dually integral) if the underlying poset has its greatest
(resp. least) element and it coincides with the neutral element of the monoid. It is not
difficult to see that ∗◦ restricted to X−(resp. X+) is integral (resp. dually integral).
3 Two new Mostert-Shields type classification theorems
In [20] the authors give a structural description of conic, involutive FLe-monoids by
proving that the cone operations of any involutive, conic FLe-monoid uniquely deter-
mine the FLe-monoid via, what is called twin rotation:
Theorem 1. [20] (Conic Representation Theorem) For any conic, involutive FLe-
monoid it holds true that
x∗◦ y=















x⊕y if x,y∈X+
x⊗y if x,y∈X−
(x→⊕y′)′if x ∈X+, y ∈X−, and x ≤y′
(y→⊗x′)′if x ∈X+, y ∈X−, and x 6≤ y′
(y→⊕x′)′if x ∈X−, y ∈X+, and x ≤y′
(x→⊗y′)′if x ∈X−, y ∈X+, and x 6≤ y′
.(1)
In [15] SIU-algebras on [0,1]have been classified. This result has been generalized in
[18], where we classify SIU-algebras over arbitrary lattices:
Theorem 2. ([18]) U=hX,∗◦,≤,t,fiis a SIU -algebra if and only if its negative cone
is a BL-algebra with components which are either product or minimum components, ⊕
is the dual of ⊗, and ∗◦ is given by (1).
Then, in [19] the authors can even weaken the quite usual continuity condition,
which was posed for the cone operators in SIU-algebras, and classify a subclass of
sharp, involutive FLe-monoids on [0,1]as follows:
Definition 2. ([19])A chain hX,≤i is called semi-separable if there exists Y⊂Xsuch
that Yis dense in Xand the cardinality of Yis smaller than the cardinality of X.
37

Definition 3. For an involutive FLe-monoid U=hX,∗◦,≤,t,fion a complete poset let
Sk(x) = max{u∈X+|u⊕x=x},if x∈X+
(inf{u∈X−|u⊗x=x})′
,if x∈X−
and call it the skeleton of ∗◦ (c.f. [21]).
Theorem 3. ([19]) On a complete, order-dense, semi-separable chain, Uis a sharp,
involutive FLe-monoid satisfying
for x ∈X−, Sk(x)′∗◦ x=x(2)
if and only if the negative cone of Uis a BL-chain without Łukasievicz components, its
positive cone is the dual of its negative cone with respect to ′, and ∗◦ is given by (1).
We remark that due to the well-known Mostert–Shields classification theorem, a BL-
chain without Łukasievicz components is exactly an ordinal sum of Boolean and prod-
uct summands in the sense of Clifford [4].
4 Applications in Substructural Logic
4.1 The logic of SIU-algebras: axiomatization and standard completeness
Substructural fuzzy logics on a countable propositional language with formulas built
inductively as usual from a set of propositional variables, binary connectives J,→,∧,
∨, and constants ⊥,⊤,f,t, with defined connectives:
¬A=de f A→f
ALB=de f ¬(¬AJ¬B)
A↔B=de f (A→B)∧(B→A)
Definition 4. MAILL (which is equivalent to FLewith ⊥and ⊤) is the substructural
logic consisting of the following axioms and rules:
(L1) A→A
(L2) (A→B)→((B→C)→(A→C))
(L3) (A→(B→C)) →(B→(A→C))
(L4) ((AJB)→C)↔(A→(B→C))
(L5) (A∧B)→A
(L6) (A∧B)→B
(L7) ((A→B)∧(A→C)) →(A→(B∧C))
(L8) A→(A∨B)
(L9) B→(A∨B)
(L10) ((A→C)∧(B→C)) →((A∨B)→C)
(L11) A↔(t→A)
(L12) ⊥ → A
(L13) A→ ⊤
A A →B
B(mp)A B
A∧B(ad j)
38

Definition 5. Uninorm logic UL and involutive uninorm logic IUL are MAILL plus
(PRL) (A→B)∧t)∨((B→A)∧t)and UL plus (INV) ¬¬A→A, respectively. Strongly
involutive uninorm logic SIUL is IUL plus f→eand (φJψ)′→(φ′Jψ′).
It turns out that SIUL is algebraizable in the sense of [1], and its equivalent algebraic
semantics is constituted by the variety of SIU-algebras.
Theorem 4. ([18]) (1) SIUL is finitely strongly complete with respect to the class of
standard SIU-algebras. (2) SIUL is not strongly complete with respect to the class of
all standard SIU-algebras. (3) Tautologicity and consequence relation in SIUL are co-
NP complete.
Acknowledgments. The first author was supported by the OTKA-76811 grant, the
SROP-4.2.1.B-10/2/KONV-2010-0002 grant, and the MC ERG grant 267589.
References
1. W. J. Blok and D. Pigozzi, Algebraizable Logics, volume 396 of Memoirs of the American
Mathematical Society, American Mathematical Society, Providence, 1989.
2. R. Cignoli, F. Esteva, L. Godo, A. Torrens, Basic Fuzzy Logic is the logic of continuous
t-norms and of their residua, Soft Computing 4 (2000), 106–112.
3. R. Cignoli, I. M. L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Rea-
soning, Kluwer, Dordrecht, 2000.
4. A. H. Clifford, Naturally totally ordered commutative semigroups, Amer. J. Math. 76 (1954),
631–646.
5. Personal communication.
6. P.T. Johnstone, Stone spaces, Cambridge University Press, Cambridge, 1982.
7. N. Galatos, P. Jipsen, T, Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at
Substructural Logics, Volume 151. Studies in Logic and the Foundations of Mathematics
(2007) 532
8. P. H´ajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
9. U. H¨ohle, Commutative residuated l-monoids, in: Topological and Algebraic Structures in
Fuzzy Sets, A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, (E.P.
Klement, S. E. Rodabaugh, eds.), Trends in Logic, vol 20. Kluwer Academic Publishers,
Dordrecht, 2003, 53–106.
10. S. Jenei, Erratum ”On the reflection invariance of residuated chains, Annals of Pure and
Applied Logic 161 (2009) 220-227”, Annals of Pure and Applied Logic 161 (2010), 1603-
1604
11. S. Jenei, On the geometry of associativity, Semigroup Forum, 74 (3): 439–466 (2007)
12. S. Jenei, Structure of Girard Monoids on [0,1], in: Topological and Algebraic Structures in
Fuzzy Sets, A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, (E.P.
Klement, S. E. Rodabaugh, eds.), Trends in Logic, vol 20. Kluwer Academic Publishers,
Dordrecht, 2003, 277-308.
13. S. Jenei, On the structure of rotation-invariant semigroups, Archive for Mathematical Logic,
42 (2003), 489–514.
14. S. Jenei, On the reflection invariance of residuated chains, Annals of Pure and Applied Logic,
161 (2009) 220–227.
39

15. S. Jenei, On the relationship between the rotation construction and ordered Abelian groups,
Fuzzy Sets and Systems, 161 (2), 277–284.
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18. S. Jenei, F. Montagna, Strongly involutive FLe-algebras, (submitted)
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20. S. Jenei, H. Ono, On involutive FLe-algebras, (submitted)
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Construction and decomposition, Fuzzy Sets and Systems, 128 (2002), 197–208.
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of Pure and Applied Algebra
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contour line, Fuzzy Sets and Systems (2007) vol. 158 (8) pp. 843–860
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111–120.
30. Ward, M. and R. P. Dilworth, Residuated lattices, Transactions of the American Mathemati-
cal Society 45: 335–354, 1939
40

Level dependent capacities and integrals
Erich Peter Klement1, Anna Koles´arov´a2, Radko Mesiar3,4, and Andrea Stup ˇnanov´a3
1Department of Knowledge-Based Mathematical Systems
Johannes Kepler University, Linz, Austria
ep.klement@jku.at
2Institute of Information Engineering, Automation and Mathematics
Slovak University of Technology, Bratislava, Slovakia
anna.kolesarova@stuba.sk
3Department of Mathematics and Descriptive Geometry
Slovak University of Technology, Bratislava, Slovakia
{radko.mesiar,andrea.stupnanova}@stuba.sk
4Institute of Theory of Information and Automation
Czech Academy of Sciences, Prague, Czech Republic
Level dependent capacities have been proposed in [1] during the 28th Linz Seminar
in 2007 (see also [2, 4]). An axiomatic approach to universal integral based on standard
capacities was given in [3]. We discuss the axiomatization of universal integrals based
on level dependent capacities.
Given a measurable space (X,A), the set of all measurable functions from Xto
[0,1]is denoted by F(X,A)
[0,1], and the set of all capacities on (X,A)by M(X,A)
1. A level
dependent capacity on (X,A)is a family (mt)t∈]0,1]of set functions mt:A→[0,1],
where each mtis a capacity on (X,A), and for the set of all level dependent capacities
on (X,A)we write M(X,A)
1. If M1= (mt,1)t∈]0,1]and M2= (mt,2)t∈]0,1]are two level
dependent capacities then we say that M1is smaller than M2(in symbols M1≤M2)
if M1,M2∈M(X,A)
1for some measurable space (X,A), and mt,1(A)≤mt,2(A)for all
t∈]0,1]and A∈A. For a fixed M∈M(X,A)
1, a function f∈F(X,A)
[0,1]is called M-A-
measurable if the function hM,f:]0,1]→[0,1]given by
hM,f(t) = mt({f≥t})
is Borel measurable. The set of all M-A-measurable functions in F(X,A)
[0,1]will be denoted
by F(X,A,M)
[0,1]. Moreover, we put
L[0,1]=[
(X,A)∈S


[
M∈M(X,A)
1
M×F(X,A,M)
[0,1]

,
where Sis the class of all measurable spaces. Similarly,we put
D[0,1]=[
(X,A)∈SM(X,A)
1×F(X,A)
[0,1].
41

Definition 1. A function L:L[0,1]→[0,1]is called a level dependent capacity-based
universal integral if the following axioms hold:
(L1) Lis nondecreasing in each component, i.e., for each measurable space (X,A), for
all level dependent capacities M1,M2∈M(X,A)
1satisfying M1≤M2, and for all
functions f1∈F(X,A,M1)
[0,1],f2∈F(X,A,M2)
[0,1]with f1≤f2we have
L(M1,f1)≤L(M2,f2),
(L2) there is a universal integral I:D[0,1]→[0,1]such that for each measurable space
(X,A), for each capacity m∈M(X,A)
1, for each f∈F(X,A,M)
[0,1], and for each level de-
pendent capacity M= (mt)t∈]0,1]∈M(X,A)
1satisfying mt=mfor all t∈[inf f,sup f]∩
]0,1]we have L(M,f) = I(m,f),
(L3) for all pairs (M1,f1),(M2,f2)∈L[0,1]with hM1,f1=hM2,f2we have
L(M1,f1) = L(M2,f2).
Observe that, because of axiom (L2), each level dependent capacity-based universal
integral Lis an extension of some universal integral I.
Remark 1. (i) The Choquet integral with respect to level dependent capacities (intro-
duced in [2], see also [1]) is a special case of Definition 1 in the sense that the
universal integral Iin axiom (L2) is the classical Choquet integral.
(ii) The Sugeno integral based on level dependent capacities (studied in [4]) is another
special case of Definition 1: here the universal integral Iin axiom (L2) is the clas-
sical Sugeno integral.
Because of axiom (L3), for each level dependent capacity-based universal integral
Land for each pair (M,f)∈L[0,1], the value L(M,f)depends only on the function hM,f
which is Borel measurable. Denote by Vthe set of all Borel measurable functions from
]0,1]to [0,1].
Theorem 1. A function L:L[0,1]→[0,1]is a level dependent capacity-based universal
integral if and only if there is a semicopula ⊗:[0,1]2→[0,1]and a function V :V→
[0,1]satisfying the following conditions:
(V1) V is nondecreasing,
(V2) V(d·1]0,c]) = c⊗d for all c,d∈[0,1],
(V3) L(M,f) = V(hM,f)for all (M,f)∈L[0,1].
Acknowledgements. The second author was supported by the grant VEGA 1/0983/12,
the third and fourth author by the project APVV-0073-10 of the Science and Tech-
nology Assistance Agency. The third author was moreover supported by the grant
P402/11/0378 of the Czech Science Foundation.
42

References
1. S. Giove, S. Greco, and B. Matarazzo, Level dependent capacities and the generalized Cho-
quet integral, Abstracts of the 28th Linz Seminar on Fuzzy Set Theory: “Fuzzy Sets, Prob-
ability, and Statistics—Gaps and Bridges” (Linz (Austria)) (D. Dubois, E. P. Klement, and
R. Mesiar, eds.), 2007, pp. 54–55.
2. S. Greco, B. Matarazzo, and S. Giove, The Choquet integral with respect to a level dependent
capacity, Fuzzy Sets and Systems 175 (2011), 1–35.
3. E. P. Klement, R. Mesiar, and E. Pap, A universal integral as common frame for Choquet and
Sugeno integral, IEEE Trans. Fuzzy Systems 18 (2010), 178–187.
4. R. Mesiar, A. Mesiarov´a-Zem´ankov´a, and K. Ahmad, Level-dependent Sugeno integral, IEEE
Trans. Fuzzy Systems 17 (2009), 167–172.
43

Globalization of Cauchy complete preordered sets
valued in a divisible quantale
Hongliang Lai and Dexue Zhang
School of Mathematics
Sichuan University, Chengdu, China
hllai@scu.edu.cn, dxzhang@scu.edu.cn
A unital quantale (Q,&)is divisible if whenever a≤bin Q, there are c1,c2∈Q
such that a=c1&b=b&c2. Given a divisible unital quantale (Q,&), it is possible to
construct different quantaloids from it. In this note, for each divisible quantale (Q,&)
we consider two special quantaloids, Qand Q.Qhas only one object which is identified
with the top element 1 ∈Q, and Q(1,1) = Qwith composition given by α◦β=α&β.
The quantaloid Qis constructed as in [3],
–objects: elements a∈Q.
–morphisms: Q(a,b) = {α∈Q:α≤a∧b}.
–composition: β◦α= (βւb)&α=β&(bցα)for all α∈Q(a,b),β∈Q(b,c).
–the unit 1aof Q(a,a)is a.
–the partial order on Q(a,b)is inherited from Q.
AQ-category Ais a set Aequipped with a map A:A×A−→ Qsuch that
(1) ∀x∈A,A(x,x) = 1;
(2) ∀x,y,z∈A,A(y,z)&A(x,y)≤A(x,z).
Q-categories are a special case of categories enriched in a monoidal closed category
[2], and have been studied both as quantitative domains [9] and as sets endowed with
fuzzy orders [1].
AQ-category Ais a set Aequipped with a map A:A×A−→ Qsatisfying:
(1) A(x,y)≤A(x,x)∧A(y,y)for all x,y∈X;
(2) A(y,z)&(A(y,y)ցA(x,y)) ≤A(x,z)for all x,y,z∈A.
Q-categories are examples of categories enriched in a bicategory [8, 10], and can be
studied as Q-subsets with quantale-valued preorders [7].
We are concerned with the relationship between the Q-categories and Q-categories.
This problem belongs to the change-base issue in the theory of enriched categories [4].
We consider three lax functors Gb,Gfand Gfrom Qto Q, given by Gbα=bցα,
Gfα=aւαand Gα= (bցα)∧(αւa)for all α∈Q(a,b). These lax functors give
rise to three functors:
–Gb:Q-Cat −→ Q-Cat, the backward globalization functor;
–Gf:Q-Cat −→ Q-Cat, the forward globalization functor;
–G:Q-Cat −→ Q-Cat, the globalization functor.
44

Theorem 1. Suppose Ais a Cauchy complete Q-category. Then both the forward glob-
alization GfAand the backward globalization GbAare Cauchy complete Q-categories.
But, whether the functor Gpreserves Cauchy completeness remains open.
References
1. Bˇelohl´avek, R.: Concept lattices and order in fuzzy logic. Annals of Pure and Applied Logic
128, 277-298 (2004)
2. Eilenberg, S., Kelly, G.M.: Closed categories. in: Eilenberg, S., Harrison, D.K. (Eds.) Pro-
ceedings of the Conference on Categorical Algebra (La Jolla, 1965), pp. 421-562. Springer,
Berlin (1966)
3. H¨ohle, U., Kubiak, T.: A non-commutative and non-idempotent theory of quantale sets.
Fuzzy Sets Syst 166, 1-43 (2011)
4. Kelly, G.M.: Basic Concepts of Enriched Category Theory. London Mathematical Society
Lecture Notes Series, vol. 64. Cambridge University Press, Cambridge (1982)
5. Lawvere, F.W.: Metric spaces, generalized logic, and closed categories, Rend. Semin. Mat.
Fis. Milano 43, 135-166 (1973)
6. Lai, H., Zhang, D.: Complete and directed complete Ω-categories. Theor. Comput. Sci. 388,
1-25 (2007)
7. Pu, Q., Zhang, D.: Preordered sets valued in a GL-monoid. Fuzzy Set Syts. 187, 1-32 (2012)
8. Stubbe, I.: Categorical structures enriched in a quantaloid: categories, distributors and func-
tors. Theory Appl. Categories 14, 1-45 (2005)
9. Wagner, K.R.: Liminf convergence in Ω-categories. Theoretical Computer Science 184, 61-
104 (1997)
10. Walters, R.F.C.: Sheaves and Cauchy-complete categories. Cah. Topologie G´eom´etrie Dif-
ferentielle Cat´egoriques 22, 1205-1223 (1981)
45

Discrete partial metric spaces
Steve Matthews1, Michael Bukatin2, and Ralph Kopperman3
1Department of Computer Science
University of Warwick, Coventry, UK
Steve.Matthews@warwick.ac.uk
2Nokia Corporation
Boston, Massachusetts, USA
bukatin@cs.brandeis.edu
3Department of Mathematics
City College, City University of New York, New York, USA
rdkcc@ccny.cuny.edu
A partial metric space is a generalisation of a metric space introducing non zero self
distance. Originally motivated by the need to model computable partially defined infor-
mation such as the asymmetric topological spaces of Scott domain theory in Computer
Science, it now falls short in an important respect. The present cost of computing infor-
mation, such as processor time or memory used, is rarely expressible in domain theory.
In contrast contemporary algorithms incorporate tight control over the cost of com-
puting resources. Complexity theory in Computer Science has dramatically advanced
through the understanding of algorithms over discrete totally defined data structures
such as directed graphs, and without the need of partially defined information. And so
we have an unfortunate longstanding separation of partial metric spaces for modelling
partially defined computable information from the highly advanced complexity theory
of algorithms for costing totally defined computable information. It is thus reasonable
to propose that a theory of cost for partial metric spaces must be possible to help bridge
the separation of domain theory and complexity theory. Today’s talk will present our
research into understanding and resolving the issues of introducing a complexity the-
ory style notion of cost to partial metric spaces. As working examples we consider the
cost of computing a double negation ¬¬ pin two-valued propositional logic, the cost
of computing negation as failure in logic programming, and a cost model for the hia-
ton time delay proposed by Wadge. The importance of our research is to keep pushing
forward from an earlier world of classical domain theory modelling computability of
partially defined information to the contemporary reality of Computer Science being
a world of dynamic, adaptive, intelligent, & biocomputing systems. ”Building better
minds together ... No challenge today is more important than creating beneficial arti-
ficial general intelligence (AGI), with broad capabilities at the human level and ulti-
mately beyond”4. Given then a fuzzy set (A,m:A→[0,1]) so useful in modelling such
sophisticated systems it is necessary to ask what is the cost of computing m(x)for any
x∈A? More precisely, how can the definition of mbe constrained to always incorporate
an appropriate notion of cost? While we are a long way from being able to answer this
fascinating question there is a relevant role model for how category theory has already
enriched computation. The introduction of monads by Moggi5to computation and later
4Open Cog Foundation opencog.org
5Notions of computation and monads, Eugenio Moggi, Information and Computation 93(1)
46

functional programming in Haskell6is being used to formalise our understanding of
how to introduce cost to partial metric spaces. Why? Functional programming offers a
λ-calculus based model of what can be defined in a logic of computation, which can
then be enriched with monads to provide a behavioural model of how efficiently a func-
tional program is being used. From this programming experience of the complexity
of computation we work to extrapolate a theory & practice of discrete partial metric
spaces.
6www.haskell.org
47

Categories isomorphic to L-fuzzy closure system spaces
Fu-Gui Shi and Bin Pang
School of Mathematics
Beijing Institute of Technology, Beijing, China
fuguishi@bit.edu.cn, pangbin1205@163.com
1 Introduction and preliminaries
One purpose of this paper is to propose a new kind of L-fuzzy closure operators which is
equivalent to L-fuzzy closure systems. Besides, some other characterizations of L-fuzzy
closure systems will be presented.
Throughout this paper, (L,∨,∧,
′)denotes a completely distributive De Morgan al-
gebra. The smallest element and the largest element in Lare denoted by ⊥and ⊤,
respectively. The set of nonzero coprimes in Lis denoted by J(L).For a,b∈L,we say
“ais wedge below b” in symbol a≺bif for every subset D⊆L,WD>bimplies a6d
for some d∈D.
For a nonempty set X,LXdenotes the set of all L-fuzzy subsets on X.The set of
nonzero coprimes in LXis denoted by J(LX).It is easy to see that J(LX)is exactly the
set of all fuzzy points xλ(λ∈J(L)).The smallest element and the largest element in
LXare denoted by ⊥and ⊤, respectively.
Definition 1 ([7]). A mapping ϕ:LX→L is called an L-fuzzy closure system on X if it
satisfies the following conditions:
(S1) ϕ(⊥) = ⊤; (S2) ϕ(V
i∈IAi)>V
i∈Iϕ(Ai).
The pair (X,ϕ)is called an L-fuzzy closure system space if ϕis an L-fuzzy closure
system on X.
A mapping f :X→Y between two L-fuzzy closure system spaces (X,ϕX)and (Y,ϕY)is
called continuous if ∀A∈LY
,ϕX(f←(A)) >ϕY(A),where f ←is defined by f ←(A)(x) =
A(f(x)) [18].
It is easy to check that L-fuzzy closure system spaces and their continuous mappings
form a category, denoted by L-FCS.
Definition 2 ([8]). A mapping τ:LX→L is called an L-fuzzy pretopology on X if it
satisfies the following conditions:
(LFPT1) τ(⊤) = ⊤; (LFPT2) τ(W
i∈IAi)>V
i∈Iτ(Ai).
The pair (X,τ)is called an L-fuzzy pretopological space if τis an L-fuzzy pretopology
on X.A mapping f :X→Y between two L-fuzzy pretopological spaces (X,τX)and
(Y,τY)is called continuous if ∀A∈LY
,τX(f←(A)) >τY(A).
It is easy to check that L-fuzzy pretopological spaces and their continuous mappings
form a category, denoted by L-FPTOP.
Theorem 1. L-FCS is isomorphic to L-FPTOP.
48

2L-fuzzy closure systems characterized by L-fuzzy closure
operators
Definition 3. An L-fuzzy closure operator onX is a mapping C:LX→LJ(LX)satisfying
the following conditions:
(C1) C(⊥)(xλ) = ⊥for any xλ∈J(LX);
(C2) C(A)(xλ) = ⊤for any xλ6A;
(C3) A 6B⇒C(A)6C(B);
(C4) C(A)(xλ) = V
xλB>AW
yµB
C(B)(yµ).
A set X equipped with an L-fuzzy closure operator C, denoted by (X,C),is called an
L-fuzzy closure space. A mapping f :X→Y between two L-fuzzy closure spaces (X,CX)
and (Y,CY)is called continuous if ∀xλ∈J(LX),∀A∈LX
,CX(A)(xλ)6CY(f→(A))( f(x)λ).
It is easy to check that L-fuzzy closure spaces and their continuous mappings form
a category, denoted by L-FC.
Theorem 2. A mapping f :X→Y between two L-fuzzy closure spaces (X,CX)and
(Y,CY)is continuous if an d only if ∀xλ∈J(LX),∀B∈LY
,CX(f←(B))(xλ)6CY(B)( f(x)λ).
Theorem 3. If ϕis an L-fuzzy closure system on X, define Cϕ:LX→LJ(LX)as follows,
∀xλ∈J(LX),∀A∈LX
,Cϕ(A)(xλ) = ^
xλB>A
ϕ(B)′
,
then Cϕis an L-fuzzy closure operator on X.
Theorem 4. If f :(X,ϕX)→(Y,ϕY)is continuous with respect to L-fuzzy closure sys-
tems ϕXand ϕY,then f :(X,CϕX)→(Y,CϕY)is continuous with respect to L-fuzzy
closure operators CϕXand CϕY.
Theorem 5. Let Cbe an L-fuzzy closure operator on X. Define ϕC:LX→L by
∀A∈LX
,ϕC(A) = ^
xλA
(C(A)(xλ))′
.
Then ϕCis an L-fuzzy closure system on X.
Theorem 6. If f :(X,CX)→(Y,CY)is continuous with respect to L-fuzzy closure op-
erators CXand CY,then f :(X,ϕCX)→(Y,ϕCY)is continuous with respect to L-fuzzy
closure systems ϕCXand ϕCY.
Theorem 7. (1)If Cis an L-fuzzy closure operator, then CϕC=C.
(2)If ϕis an L-fuzzy closure system, then ϕCϕ=ϕ.
Theorem 8. L-FCS is isomorphic to L-FC.
49

3 The other characterizations of L-fuzzy closure systems
Definition 4. An L-fuzzy interior operator on X is a mapping I:LX→LJ(LX)satisfying
the following conditions:
(I1) I(⊤)(xλ) = ⊤for any xλ∈J(LX);
(I2) I(A)(xλ) = ⊥for any xλA;
(I3) A 6B⇒I(A)6I(B);
(I4) I(A)(xλ) = W
xλ6B6AV
yµ≺B
I(B)(yµ).
A set X equipped with an L-fuzzy interior operator I, denoted by (X,I),is called
an L-fuzzy interior space. A mapping f :X→Y between two L-fuzzy interior spaces
(X,IX)and (Y,IY)is called continuous if ∀xλ∈J(LX),∀B∈LY
,IX(f←(B))(xλ)>
IY(B)( f(x)λ).
It is easy to check that L-fuzzy interior spaces and their continuous mappings form
a category, denoted by L-FI.
Definition 5. An L-fuzzy neighborhood system on X is defined to be a set N ={Nxλ|xλ∈
J(LX)}of mappings Nxλ:LX→L satisfying the following conditions:
(LN1) Nxλ(⊤) = ⊤,Nxλ(⊥) = ⊥;
(LN2) Nxλ(A) = ⊥for any xλA;
(LN3) A 6B⇒Nxλ(A)6Nxλ(B);
(LN4) Nxλ(A) = W
xλ6B6AV
yµ≺BNyµ(B).
A set X equipped with an L-fuzzy neighborhood system N ={Nxλ|xλ∈J(LX)}, denoted
by (X,N),is called an L-fuzzy neighborhood space. A mapping f :X→Y between
two L-fuzzy neighborhood spaces (X,NX)and (Y,NY)is called continuous if ∀xλ∈
J(LX),∀B∈LY
,(NX)xλ(f←(B)) >(NY)f(x)λ(B).
The category of L-fuzzy neighborhood spaces with their continuous mappings is
denoted by L-FN.
Definition 6. An L-fuzzy quasi-coincident neighborhood system on X is defined to be a
set Q ={Qxλ|xλ∈J(LX)}of mappings Qxλ:LX→L satisfying the following condi-
tions:
(QN1) Qxλ(⊥) = ⊥,Qxλ(⊤) = ⊤;
(QN2) Qxλ(A)6=⊥ ⇒ xλA′;
(QN3) A 6B⇒Qxλ(A)6Qxλ(B);
(QN4) Qxλ(A) = W
xλB>A′V
yµB
Qyµ(B′).
A set X equipped with an L-fuzzy quasi-coincident neighborhood system Q ={Qxλ|xλ∈
J(LX)}, denoted by (X,Q),is ca lled an L-fuzzy quasi-coincident neighborhood space. A
mapping f :X→Y between two L-fuzzy quasi-coincident neighborhood spaces (X,QX)
and (Y,QY)is called continuous if ∀xλ∈J(LX),∀B∈LY
,(QX)xλ(f←(B)) >(QY)f(x)λ(B).
The category of L-fuzzy quasi-coincident neighborhood spaces with their continu-
ous mappings is denoted by L-FQN.
Theorem 9. L-FCS, L-FPTOP, L-FC, L-FI, L-FN and L-FQN are all isomorphic.
50

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51

Variable range categories of approximate systems
Alexander ˇ
Sostak1,2
1Department of Mathematics
University of Latvia, Riga, Latvia
2Institute of Mathematics and Computer Science
University of Latvia, Riga, Latvia
sostaks@latnet.lv, sostaks@lanet.lv
1 Introduction and motivation
In our paper [14] the concept of an M-approximate system where Mis a fixed complete
lattice was introduced and basic properties of the category of M-approximate systems
were studied. We regard the concept of an M-approximate system and the correspond-
ing category as the framework for a unified approach to various categories related to
(fuzzy) (bi)toplogical spaces ([2], [5], [3], [4], [13], [8], [11], [12], etc) and to (fuzzy)
rough sets ([10], [1], etc). Although the attempts to study the relations between fuzzy
topological space and fuzzy rough sets and to introduce a context allowing to give a
unified view on these notions were undertaken also by other authors, see e.g. [6], [7],
[15], [16], the approach presented in [14] is essentially different. In this work we con-
tinue the research of M-approximate systems. However, as different from our previous
work here we consider the case of a variable range M, that is allow to change lattice
M. In particular this alllows to include also the category of LM-topological spaces with
varied lattice Min the scope of our research. In our work two lattices will play the fun-
damental role. The first one is an infinitely distributive lattice, that is a complete lattice
L= (L,≤,∧,∨),satisfying the infinite distributivity laws a∧(Wi∈Ibi) = Wi∈I(a∧bi)
and a∨(Vi∈Ibi) = Vi∈I(a∨bi)for all a∈L,{bi|i∈I} ⊆ L.Its top and bottom ele-
ments are 1Land 0Lrespectively. Sometimes we assume that the lattice Lis equipped
with an order reversing involution c:L→L. In particular, if Lis enriched with a bi-
nary operation L∗L→Lsuch that L= (L,≤,∧,∨,∗),is Girard monoid, in particular
an MV-algebra then involution is naturally defined by ac= (a7→ 0)7→ 0.The second
lattice belonging to the context of our work is M. At the moment we assume only its
completeness, however in applications to LM-fuzzy topology we need to assume that
it is complete distributive. Its bottom and top elements are 0Mand 1Mresp., 0M6=1M.
that is Mcontains at least two elements. For the categories of complete lattices, com-
plete infinitely distributive lattices and of complete infinitely distributive lattices with
an order reversing involution will be denoted CLAT, IDL and IDLC respectively.
2 Basic definitions
Definition 1. An upper M-approximate operator on Lis a mapping u :L×M→Ls.t.
(1u) u(0L,α) = 0L∀α∈M;
52

(2u) a≤u(a,α)∀a∈L,∀α∈M;
(3u) u(a∨b,α) = u(a,α)∨u(b,α)∀a,b∈L,∀α∈M;
(4u) u(u(a,α),α) = u(a,α)∀a∈L,∀α∈M;
(5u) α≤β,α,β∈M=⇒u(a,α)≤u(a,β)∀a∈L.
Definition 2. A lower M-approximate operator on Lis a mapping l :L×M→Ls. t.
(1l) l(1L,α) = 1L∀α∈M;
(2l) a≥l(a,α)∀a∈L,∀α∈M;
(3l) l(a∧b,α) = l(a,α)∧l(b,α)∀a,b∈L,∀α∈M;
(4l) l(l(a,α),α) = l(a,α)∀a∈L,∀α∈M;
(5l) α≤β,α,β∈M=⇒l(a,α)≥l(a,β)∀a∈L.
Definition 3. A quadraple (L,M,u,l), where u :L×M→Land l :L×M→Lare
upper and lower M-approximate operators on L, is called an M-approximate system
on Lor just an approximate system. An approximate system is called
(T) tight, if u(a,0M) = l(a,0M) = a∀a∈L;
(SA) semicontinuous from above if
u(a,Vi∈Iαi) = Vi∈Iu(a,αi),l(a,Vi∈Iαi) = Wi∈Il(a,αi);
(WA) weakly semicontinuous from above if
u(a,αi) = a∀i∈I=⇒u(a,Vi∈Iαi) = a and l(a,αi) = a∀i∈I=⇒l(a,Wi∈Iαi) = a.
If X is a set, Lis a lattice, L=LXand (L,M,u,l)is an approximate system, the
tuple (X,L,M,u,l)is called an approximate space.
3 Lattice of M-approximate systems on a fixed lattice L
Let A S M(L)stand for the family of all M-approximate systems (L,M,u,l)where L
and Mare fixed. Further, let T-AS M(L), D-A S M(L), SA-A S M(L), WA-A S M(L)stand
on the subfamilies of A S M(L)consisting respectively of tight, self-dual, semicontinu-
ous from above, and weakly semicontinuous from above M-approximate systems on L,
respectively. We introduce a partial order on the family A S M(L)by setting
(L,u1,l1)(L,u2,l2)iff u1≥u2and l1≤l2.
Theorem 1. (A S M(L),)is a complete lattice. Its top element is (L,u⊤,l⊤)where
u⊤(a,α) = l⊤(a,α) = a for every a ∈Land every α∈Mand its bottom element is
(L,u⊥,l⊥)where
u⊥(a,α) = 1Lif a 6=0L
0L,if a =0L
l⊥(a,α) = 0Lif a 6=1L
1L,if a =1L
53

Theorem 2. (T-A S M(L),)is a complete lattice whose top element is the same as in
(A S M(L),), that is (L,M,u⊤,l⊤), and whose bottom element is (L,M,ut
⊥,lt
⊥), where
ut
⊥(a,α) = 


1Lif a 6=0Land α6=0M
0L,if a =0L
a,if α=0M
lt
⊥(a,α) = 


0Lif a 6=1Land α6=0M
1L,if a =1L
a,if α=0M
Theorem 3. The family (WA-AS M(L),)of weakly semicontinuous from above M-
approximate systems is a complete sublattice of the lattice (A S M(L),).
Theorem 4. Let D∈Ob(IDLC).Then the family D-A S M(L),)of self-dual approx-
imate systems is a complete sublattice of the lattice (A S M(L),).
4 Category AS of approximate systems
Let AS be the family of all approximate systems (L,M,u,l). To consider AS as a cat-
egory whose class of objects are all M-approximate systems (L,M,u,l)where L∈
Ob(IDL)and M∈Ob(CLAT)we have to specify its morphisms. Given two approxi-
mate systems (L1,M1,u1,l1),(L2,M2,u2,l2)∈Ob(AS)by a morphism
F:(L1,M1,u1,l1)→(L2,M2,u2,l2)
we call a pair F= ( f,ϕ)such that
(1m) f:L1→L2is a morphism in the category IDLop;
(2m) ϕ:M1→M2is a morphism in the category CLATop;
(3m) u1(f(b),ϕ(β)) ≤f(u2(b,β)) ∀b∈L2,∀β∈M2;
(4m) f(l2(b,β)) ≤l1(f(b),ϕ(β)) ∀b∈L2,∀β∈M2
Remark 1. The category ASM, where Mis a fixed lattice, which was studied in [14]
can be identified with a subcategory of the category T-AS having M-approximate sys-
tems (L,M,u,l)as objects and pairs F= ( f,idM):(L1,M1,u1,l1)→(L2,M,u2,l2)as
morphisms. (idM:M→Mstands for an identity mapping.) In particular, in case when
Mis a two-point lattice we obtain the category AS2.
Theorem 5. Every source Fi:(L1,M1)→(Li,Mi,ui,li),i∈Iin AS has a unique
initial lift Fi:(L1,M1,u1,l1)→(Li,Mi,ui,li),i∈I.
Theorem 6. Every sink Fi:(Li,Mi,ui,li)→(L1,M1),i∈Iin AS has a unique final
lift Fi:(Li,Mi,ui,li)→(L1,M1,u1,l1),i∈I
Corollary 1. Category AS is topological over the category IDLo p ×CLATo p with re-
spect to the forgetful functor F:AS −→ IDLop ×CLATo p
.
54

We study also the categorical properties of the full subcategories of AS whose ob-
jects are tight, self-dual, and (weakly) semicontinuous from above approximate systems
as well as some other classes of approximate systems. In particular, we show that
Theorem 7. Category D-AS of self-dual approximate systems is topological over the
category IDLCop ×CLATop with respect to the forgetful functor F: D-AS→IDLCo p ×
CLATo p
.
Some subcategories of AS determined by restricted classes of morphisms will be
also in the scope of our interest. Finally we will discuss different concrete categories
related to fuzzy (bi)topology and fuzzy rough sets regarded as subcategories of AS.
Acknowledgments. The author gratefully acknowledges a partial financial support by
the LZP (Latvian Science Council) research project 09.1570, as well as a partial finan-
cial support by the ESF research project 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008.
References
1. P. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Internat. J. General Systems
17 (1990), 191–209.
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3. U. H¨ohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78
(1980) 659-673.
4. U. H¨ohle, A. ˇ
Sostak, Axiomatics for fixed-based fuzzy topologies, Chapter 3 in [9]
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6. J. Kortelainen, On relationship between modified sets, topological spaces and rough sets,
Fuzzy Sets and Syst. 61 (1994), 91-95.
7. J. J¨arvinen, J. Kortelainen, A unified study between modal-like operators, Topologies
and Fuzzy Sets, Turku Centre for Computer Sciences, Turku, Finland, Technical Reports
642, 2004.
8. T. Kubiak, A.ˇ
Sostak, Foundations of the theory of (L,M)-fuzzy topological spaces, 30th
Linz Seminar on Fuzzy Sers, Linz, Austria, February 2009. Abstracts,70–73.
9. Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory , U. H¨ohle, S.E.
Rodabaugh eds. - Handbook Series, vol.3, Kluwer Acad. Publ., - 1999.
10. Z. Pawlak, Rough sets, International J. of Computer and Information Sciences, 11 (1982),
341-356.
11. S.E. Rodabaugh, A categorical accomodation of various notions of fuzzy topology, Fuzzy
Sets and Syst.,9(1983), 241–265.
12. S.E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, Chapter 4 in
[9].
13. A. ˇ
Sostak, Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), 662–701.
14. A. ˇ
Sostak Towards the theory of M-approximate systems: Fundamentals and examples,
Fuzzy Sets and Syst., 161 (2010), pp.2440-2461.
15. Y.Y. Yao, A comparative study of fuzzy sets and rough sets, Inf. Sci., 109 (1998), 227–242.
16. Y.Y. Yao, On generalizing Pawlak approximation operators, Proc. of the First Internat
Conf ”Rough Sets and Current Trends in Computing” (1998), 298–307.
55

Category Theory in Statistical Learning?
Milan Stehl´ık
Department of Applied Statistics
Johannes Kepler University, Linz, Austria
Milan.Stehlik@jku.at
David Corfield was asked recently by someone for his opinion on the possibility
that category theory might prove useful in machine learning. First of all, he would not
want to give the impression that there are signs of any imminent breakthrough. For
other areas of computer science the task would be easier (nowadays!). Category theory
features prominently in theoretical computer science as described in books such as [2].
And what about statistics? One direct help may be a probability theory. In a cou-
ple of web posts Corfield discussed a construction of probability theory in terms of a
monad. He pointed out a natural inclination of the Bayesian to think about distributions
over distributions fits this construction well.
Moreover, Graphical models, which include directed graphs, together with Bayesian
networks, may sometimes form a symmetric monoidal category.
Another dimension to spaces of probability distributions is that they can be studied
by differential geometry in a field known as information geometry. For an insightful
treatment in the context of nonlinear models see [5], general treatment may be found in
[1].Beside the above mentioned issues, one practical application for empirical statis-
tics, the “categorization” of inference function will be discussed. In [3] we have real-
ized (by empirical research) a need of non-crisp monotonicity for Fisher information
of experiments under heteroscedasticity. The classical Fisher information is based on
the “classical” score function, used by the pioneers of modern statistics (Karl Pearson,
Francis Y. Edgeworth and Sir Ronald A. Fisher) have been introduced as a local change
of log-likelihood w.r.t. to a parameter of interest, more less in case to case studies.
However, an alternative score can be defined ([4]) and proven to have some desirable
properties ([6] and [7]) in classical statistical inference. In nonparametrics, a similar
inference function, so called influence function is used. A practical discussion of this
aspects in a context of the “categorization” of inference function will be given.
References
1. Amari S., Chapter 2: Differential Geometrical Theory of Statistics, Source: S.-I. Amari, O.
E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, and C. R. Rao Differential geometry in
statistical inference: (Hayward, CA: Institute of Mathematical Statistics, 1987), 19-94.
2. Barr M. and Wells Ch. (1999). Category Theory for Computing Science, Centre de
recherches math´ematiques
56

3. Boukouvalas A., Cornford D. and Stehl´ık M. (2009). Approximately Optimal Experimental
Design for Heteroscedastic Gaussian Process Models, Technical Report of Aston University,
UK.
4. Fabi´an Z. (2001). Induced cores and their use in robust parametric estimation, Communica-
tion in Statistics, Theory Methods, 30 , pp.537-556.
5. P´azman A. (1993). Nonlinear statistical Models, Kluwer Acad. Publ., Dordrecht, chapters
9.1, 9.2 and 9.5
6. Stehl´ık M., Potock´y R., Waldl H. and Fabi´an Z. (2010). On the favorable estimation for
fitting heavy tailed data, Computational Statistics, 25:485-503.
7. Stehl´ık M., Fabi´an Z. and Stˇrelec L. (2011). Small sample robust testing for Normality
against Pareto tails, Communications in Statistics - Simulation and Computation (in press).
57

Quantaloid-enriched categories
for multi-valued logic
and other purposes
Isar Stubbe
Laboratoire de Mathmatiques Pures et Appliques
Universit du Littoral-Cˆote d’Opale, Calais, France
isar.stubbe@lmpa.univ-littoral.fr
In this lecture I shall aim to give an overview of the basic concepts in the theory of
quantaloid-enriched categories, giving as many examples as time permits. First I shall
recall what quantales and quantaloids are, and how one computes extensions and lift-
ings in them. Then I shall define categories, functors and distributors enriched in a quan-
taloid, saying something about the universal property of quantaloid-enrichment too. I
shall explain how every functor between quantaloid-enriched categories determines a
left adjoint distributor, and that this very fact is at the heart of quantaloid-enriched cat-
egory theory.By way of illustration I shall show how to define adjunctions, presheaves,
(co)limits, (co)completions, and so on. Further I shall say a word about the symmetri-
sation of quantaloid-enriched categories. And finally I shall indicate the link between
quantaloid-enriched categories on the one hand, and modules on a quantaloid on the
other. This lecture should provide (more than) the background that is needed for H.
Heymans’ lecture on sheaf theory via quantaloid-enrichment.
58

On the characterisation of
regular left-continuous t-norms
Thomas Vetterlein
Department of Knowledge-Based Mathematical Systems
Johannes Kepler University, Linz, Austria
Thomas.Vetterlein@jku.at
1 Introduction
Quantales are complete lattices endowed with an associative binary operation ⊙dis-
tributing from both sides over arbitrary joins [Ros]. A quantale is called strictly two-
sided if there is a top element that is neutral w.r.t. ⊙, and it is called commutative if
⊙is commutative. In the special case that the complete lattice is the real unit interval
endowed with the natural order, a strictly two-sided, commutative quantale is an algebra
well-known in fuzzy logic: a left-continous (l.-c.) t-norm algebra [KMP].
We consider this type of structure from a constructive point of view, being interested
in its complete description. Our viewpoint is algebraic as we classify l.-c. t-norm alge-
bras up to isomorphism only. However, we also make use of methods from analysis, in
particular from the theory of functional algebras.
Let ([0,1];≤,⊙,0,1)be a l.-c. t-norm algebra. We denote by (Λ⊙;≤,◦,¯
0,id)the
associated translation tomonoid. That is, Λ⊙consists of all (inner right) translations
λ⊙
a:[0,1]→[0,1]:x7→ x⊙a
by some a∈[0,1];≤is the pointwise order; ◦is the function composition; and ¯
0 is the
zero constant function, id the identical function. The isomorphism a7→ λ⊙
aof the semi-
group ([0,1];⊙)and its translation semigroup (Λ⊙;◦)[ClPr] extends to an isomorphism
between ([0,1];≤,⊙,0,1)and (Λ⊙;≤,◦,¯
0,id). We have [Vet]:
Theorem 1. Let ⊙be a l.-c. t-norm. Then Λ⊙is a set of functions from [0,1]to [0,1]
with the following properties:
(T1) Every f is increasing.
(T2) Every f and g commute.
(T3) For every t ∈[0,1], there is exactly one f such that f (1) = t.
(T4) Every f is left-continuous.
Conversely, let Λbe a set of functions from [0,1]to [0,1]fulfilling (T1)–(T4). Then there
is a unique l.-c. t-norm ⊙such that Λ=Λ⊙.
The following heuristic argument may illustrate how the present work was moti-
vated. Consider the following depictions of the translation tomonoids of the three basic
continuous t-norms:
59

0
0.2
0.4
0.6
0.8
1
0 0.2 0.40.60.81
Łukasiewicz t-norm
0
0.2
0.4
0.6
0.8
1
0 0.2 0.40.60.81
Product t-norm
0
0.2
0.4
0.6
0.8
1
0 0.2 0.40.60.81
G¨odel algebra
Observe that, in each of these cases, if the picture was almost completely covered
and we were able to inspect an arbitrarily narrow stripe below the identity line only,we
would be able to reconstruct the whole functional algebra. In fact, the functions in a
neighborhood of the identity either generate the whole algebra, or it can be concluded
that all functions are idempotent and thus uniquely determined by the intersection of
their graphs with the identity line.
2 Regular l.-c. t-norms: the simple case
The exact facts have been examined in the paper [Vet], of which the present work is
the continuation. As might be expected, the above observations do not apply for all l.-c.
t-norm algebras. We restrict our attention to the following subclass.
Definition 1. A l.-c. t-norm ⊙is called regular if the following conditions hold:
(1) There is an n <ωsuch that each f ∈Λ⊙has at most n discontinuity points.
(2) For t ∈[0,1], put e(t) = inf {s:s⊙t=t}. Then there are 0=v0<v1< . . . < vk=1
such that for each i =0,..., k−1, the map e|(vi,vi+1)is continuous and one of the
following possibilities holds:
(a) e|(vi,vi+1)is constant r , and we have r ⊙t=t for all t ∈(vi,vi+1);
(b) e|(vi,vi+1)is strictly monotonous.
Even if this condition looks special, the class of t-norm algebras based on regular
l.-c. t-norm is not neglible – in the sense that it generates the whole variety of MTL-
algebras.
Regular l.-c. t-norm algebras can be decomposed in a specific way. Namely, let
(Λ⊙;≤,◦,
0
,
id
)be the translation tomonoid of the regular l.-c. t-norm ⊙. Then we may
determine a characteristic sequence of points (v0,...,vk)– cf. the definition of regular-
ity –, called a frame for ⊙. For each basic interval (vi,vi+1], we consider the induced
translation tomonoid: Λ(vi,vi+1]={f(vi,vi+1]:f∈Λ⊙},
where f(vi,vi+1]:(vi,vi+1]→(vi,vi+1]:a7→ f(a)∨vi. We call (Λ(vi,vi+1];≤,◦,
0
,
id
)a
basic tomonoid of ⊙.
Theorem 2. Any basic tomonoid associated to some l.-c. t-norm belongs to one out of
six isomorphism classes.
60

3 Regular l.-c. t-norms: the general case
Knowing the basic tomonoids associated to a l.-c. t-norm ⊙means to know how the
translations by the elements of each basic interval act on this same interval. This knowl-
edge alone may or may not determine the whole t-norm algebra. In the former case, a
l.-c. t-norm is fully characterised by (1) the size kof a frame, (2) the type of each of the
kbasic tomonoids, and (3) the intervals parametrising the basic tomonoids.
The question how the translations by the elements of one basic interval act on the
remaining intervals has not yet been addressed; this is the topic of the present work.
Let (Λ⊙;≤,◦,
0
,
id
)be the translation tomonoid of the regular l.-c. t-norm ⊙. Let
(v0,...,vk)be a frame for ⊙. For each pair of distinct intervals (vi,vi+1]and (vj,vj+1],
put
H(vj,vj+1]
(vi,vi+1]={f(vj,vj+1]
(vi,vi+1]:f∈Λ⊙},
where f(vj,vj+1]
(vi,vi+1]:(vi,vi+1]→(vj,vj+1]:a7→ (f(a)∨vj)∧vj+1. We call H(vj,vj+1]
(vi,vi+1]a
lower tomonoid of ⊙.
It turns out that the lower tomonoids are largely determined by the basic tomonoids:
for each i,j, the algebra H(vj,vj+1]
(vi,vi+1]is determined as follows. There is a totally ordered
set of functions H, uniquely determined by Λ(vi,vi+1]and Λ(vj,vj+1], such that H(vj,vj+1]
(vi,vi+1]
is an interval of H. As a consequence, for the description of a general regular t-norm,
we need in addition to (1), (2), and (3) above to specify (4) the relevant intervals of the
lower tomonoids, and (5) the intervals parametrising the lower tomonoids.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
The three-part H´ajek t-norm
1
0.9
0.75
0.6
0.45
0.3
0.1
61

As an example, we consider a t-norm that was proposed in a modified form by P.
H´ajek [Haj]:
a⊙b=















a(3b−2)if a≤1
3and b>2
3,
3ab −2a−b+1 if 1
3<a≤2
3and b>2
3,
3ab −2a−2b+2 if a,b>2
3,
0 if a≤1
3and b≤2
3,
3ab −a−b+1
3if 1
3<a,b≤2
3
for a,b∈[0,1]. We have the following characteristic data. (1) Size of frame: 3. (2) Type
of basic tomonoids: product; product; product. (3) Parametrising intervals: [2
3,1);[2
3,1);
[2
3,1). (4) Intervals of the lower algebras: full; full; full. (5) Intervals parametrising the
lower algebras: [1
3,2
3);[1
3,2
3);[0,1
3).
4 Conclusion
We have shown that any left-continous t-norm fulfilling the condition of regularity al-
lows a particular type of decomposition into finitely many constituents. Namely, the
real unit interval may be divided into finitely many subintervals and the tomonoids of
translations by the elements of one interval restricted to another interval may be indi-
cated by means of six isomorphism classes. In short, we may associate to a regular l.-c.
t-norm its characteristic data, describing how the t-norm is composed from a finite set
of specific constituents.
Conversely, it is not difficult to check if given data to construct a l.-c. t-norm is
actually the characteristic data of a l.-c. t-norm. An easy criterion to decide this question
is, however, not known to us.
References
[ClPr] A. H. Clifford, G. B. Preston, “The algebraic theory of semigroups”, vol. 1, American
Mathematical Society, Providence 1961.
[Haj] P. H´ajek, Observations on the monoidal t-norm logic,
Fuzzy Sets Syst.
132 (2003), 107 -
112.
[KMP] E. P. Klement, R. Mesiar, E. Pap, “Triangular Norms”, Kluwer Acad. Publ., Dordrecht
2000.
[Ros] K. I. Rosenthal, Quantales and their applications, Longman Scientific & Technical, Essex
1990.
[Vet] T. Vetterlein, Regular left-continuous t-norms,
Semigroup Forum
77 (2008), 339 - 379.
62

Type-2 operations on finite chains
Carol Walker and Elbert Walker
Department of Mathematical Sciences
New Mexico State University, Las Cruces, New Mexico, USA
{hardy, elbert}@nmsu.edu
1 Introduction
The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit
interval into itself, with operations certain convolutions of these mappings with respect
to pointwise max and min. This algebra has been studied extensively as indicated in the
references below. The basic theory depends on the fact that [0,1]is a complete chain,
so lends itself to various generalizations and consideration of special cases. This paper
develops the theory where each copy of the unit interval is replaced by a finite chain.
Most of the theory goes through, but there are several special circumstances of interest.
2 The Algebra M(mn)
For a positive integer n, let nbe the set {1,2,...,n}. This set has its usual linear order
which we denote by ≤, max and min operations denoted ∨and ∧, negation given by
¬k=n−k+1,and the obvious constants 1 and n. With these operations, nbecomes a
De Morgan algebra, in fact a Kleene algebra since it also satisfies a∧ ¬a≤b∨ ¬b.
We denote by mnthe set {f:n→m}of all mappings from the set ninto the set
m. The algebra M(mn)consists of the set mnwith operations given in the following
definition.
Definition 1. On mn,let
(f⊔g)(i) = _
j∨k=i
(f(j)∧g(k))
(f⊓g)(i) = _
j∧k=i
(f(j)∧g(k))
¬f(i) = _
j=¬i
f(j) = f(¬i)
¯
1(i) = m if i =m
1if i 6=mand ¯
0(i) = m if i =1
1if i 6=1
Thus we have the algebra
M(mn) = (mn
,⊔,⊓,¬,¯
0,¯
1)
63

There are two other operations on the functions in mn, namely pointwise max and
min. We also denote these by ∨and ∧, respectively. Just as in the case M([0,1][0,1]),
these operations help in determining the properties of the algebra M(mn)via the auxil-
iary operations fL(i) = ∨j≤if(j)and fR(i) = ∨j≥if(j).
The operations ⊔and ⊓in M(mn)can be expressed in terms of the pointwise max
and min of functions in two different ways, as follows.
Theorem 1. The following hold for all f ,g∈M(mn).
f⊔g=f∧gL∨fL∧g= ( f∨g)∧fL∧gL
f⊓g=f∧gR∨fR∧g= ( f∨g)∧fR∧gR
Using these auxiliary operations, it is fairly routine to verify the following properties
of the algebra M(mn). The details of the proofs are almost exactly the same as for the
algebra M([0,1][0,1]), which are given for example in [9].
Corollary 1. Let f , g, h ∈M(mn). Some basic equations follow.
1. f ⊔f=f ; f ⊓f=f
2. f ⊔g=g⊔f ; f ⊓g=g⊓f
3. f ⊔(g⊔h) = ( f⊔g)⊔h; f ⊓(g⊓h) = ( f⊓g)⊓h
4. f ⊔(f⊓g) = f⊓(f⊔g)
5. ¯
1⊓f=f; ¯
0⊔f=f
6. ¬¬ f=f
7. ¬(f⊔g) = ¬f⊓ ¬g; ¬(f⊓g) = ¬f⊔ ¬g
The elements of M(mn)may be deonoted by n-tuples (a1,a2,...,an)of elements
of m. Note that with this notation, in mnthe element ¯
1 is (1,1, . . . , 1,m)and ¯
0 is
(m,1,1,...,1). Further note that the algebra has an absorbing element (1,1,...,1). Fi-
nally, ¬(a1,a2,...,an) = (an,an−1,...,a1).
3 The Main Results
Each of ⊔and ⊓, being idempotent, commutative and associative, gives rise to a partial
order. These partial orders are defined by f≤⊔gif f⊔g=gand f≤⊓gif f⊓g=f.
Theorem 2. The partial orders ≤⊔and ≤⊓are lattice orders.
The equations listed above do not form an equational basis for M(mn). We do not
know an equational basis for M(mn)nor even if a finite one exists. However, similar to
the case of M([0,1][0,1])[4], we do get the following.
Theorem 3. For m ≥2, the algebras M(mn)and M(2n)generate the same variety and
thus satisfy the same equations.
Theorem 4. Let n≥5.Then M(2n)and M(25)generate the same variety and thus
satisfy the same equations.
64

One main objective of this paper was to show that the automorphism group of the
retract (mn
,⊔,⊓)of M(mn)is trivial, that is, has only one element. To effect this, the
irreducible elements of (mn
,⊔,⊓)were determined.
Theorem 5. Let m,n≥2. The irreducible elements of (mn
,⊔,⊓)are these.
1. The absorbing element (1,1,...,1).
2. The n-tuple with miin the i-th place and 1elsewhere.
3. The element m1∨mn.
4. If n =2, n-tuples that contain m, and the absorbing element.
Using the theorem above and long sequence of lemmas, we get the following.
Theorem 6. The automorphism group of (mn
,⊔,⊓)has only one element.
4 Comments
One principal result of this paper is that the partial order given by the operation ⊔is a
lattice, and analogously for ⊓. For the operation ⊔, the sup of two elements fand gis
f⊔g, but the inf of the two elements is the sup of the set of all elements below both f
and g. The elements fand gare n-tuples of elements of m, and the inf is given by some
formula in the elements in these two n-tuples.
Problem 1. Find a formula for the inf of two elements in the lattice determined by ⊔.
And similarly, do the same for the lattice determined by ⊓.
Problem 2. In the case of the algebra ([0,1][0,1]
,⊔), determine whether or not the partial
order determined by ⊔is a lattice.
In the case of 23, the lattices determined by ⊔and by ⊓are both distributive, but
this is not true for all mn.
Problem 3. For which mnare the lattices determined by ⊔and ⊓distributive? We con-
jecture none for mand n≥3.
The proof that Aut(mn
,⊔,⊓)consists of only the identity automorphism was ef-
fected by a long sequence of lemmas, etc. Hopefully, there is a much shorter and less
computational proof.
Problem 4. Find a proof that Aut(mn
,⊔,⊓)is trivial that is more conceptual, less com-
putational, and shorter.
In showing that the automorphism group of (mn
,⊔,⊓)consists of only the identity
automorphism, we used in the proof that an automorphism preserved both ⊔and ⊓.
But small examples show that the automorphism group of (mn
,⊔)is just the identity
automorphism, and we suspect that this is true in general, but have no proof.
Problem 5. Find the automorphism group of (mn
,⊔).(Since (mn
,⊔)and (mn
,⊓)are
isomorphic, their automorphism groups will be isomorphic.)
65

Finally, there are many ways to specialize and to generalize the truth-value algebra
([0,1][0,1]
,⊔,⊓,¬,¯
0,¯
1)of type-2 fuzzy sets. We have just taken a finite chain for each
interval [0,1]. For example, one could take any two completelattices instead, or substi-
tute one finite chain for one of the intervals [0,1], and so on. Such investigations may
be of interest.
References
1. D. Dubois, H. Prade, Operations in a fuzzy-valued logic, Information and Control,43 (1979)
224-240.
2. M. Gehrke, C. Walker, E. Walker, De Morgan systems on the unit interval, International
Journal of Intelligent Systems, 11(1996) 733-750.
3. J. Harding, C. Walker, E. Walker, Lattices of Convex Normal Functions, Fuzzy Sets and
Systems,159 (2008), 1061-1071.
4. J. Harding, C. Walker, E. Walker, The variety generated by the truth-value algebra of type-2
fuzzy sets, Fuzzy Sets and Systems,161(5) (2010) 735-749.
5. J. Harding, C. Walker, E. Walker, Convex normal functions revisited, Fuzzy Sets and Systems,
161 (2010) pp. 1343-1349.
6. J. Harding, C. Walker, E. Walker, Projective Bichains, Algebra Universalis, to appear
7. J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems. Prentice Hall PTR, Upper Saddle
River, NJ, (2001)
8. M. Mizumoto K. Tanaka, Some Properties of Fuzzy Sets of Type-2, Information and Control,
31 (1976) 312-340.
9. C. Walker E. Walker, The Algebra of Fuzzy Truth Values, Fuzzy Sets and Systems,149 (2005)
309-347.
10. C. Walker E. Walker, Automorphisms of Algebras of Fuzzy Truth Values, International Jour-
nal of Uncertainty, Fuzziness and Knowledge-based Systems,14(6) (2006) 711-732.
11. C. Walker, E. Walker, Automorphisms of Algebras of Fuzzy Truth Values II, International
Journal of Uncertainty, Fuzziness and Knowledge-based Systems,16(5) (2008) 627-643.
12. C. Walker, E. Walker, Sets With Type-2 Operations, International Journal of Approximate
Reasoning, 50 (2009) 63-71.
13. L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,
Inform Sci.,8(1975) 199-249
66

Convergence and compactness in fuzzy metric spaces
Yueli Yue
Department of Mathematics
Ocean University of China, Qingdao, China
ylyue@ouc.edu.cn
Abstract. As H¨ohle observed in [9] the fact that the topology associated to a
probabilistic metric space is metrizable means that,from this topological point
of view, probabilistic metric spaces are always equivalent to ordinary metric
spaces,and the problem of topologization of probabilistic metric spaces is not
satisfactorily solved. He proposed many-valued topologies as suitable tools for
this purpose. Hence in [15], we endowed George and Veeramani’s fuzzy metric
(which has close relation to probabilistic metric) with many-valued structures-
fuzzifying topology and fuzzifying uniformity. The aim of this paper is to go on
studying the properties of George and Veeramani’s fuzzy metric. We will give the
concept of convergence degree and generalize the convergence and compactness
theories in metric spaces to Veeramani’s fuzzy metric spaces.
1 Introduction
Metric space plays an important role in the research and applications of topology. Con-
vergence theory is an another important part in metric spaces and is the key tool in
studying completeness. Probabilistic metric space, a generalization of the ordinary met-
ric space, was first studied by Menger [12] and further developed by Schweizer and
Sklar [14]. Inspired by the notion of probabilistic metric spaces, Kramosil and Michalek
[10] in 1975 introduced the notion of fuzzy metric, a fuzzy set in the Cartesian product
X×X×ℜsatisfying certain conditions (see Definition 2.12 for a similar form). George
and Veeramani [1–3] slightly modified the definition of Kramosil and Michalek’s fuzzy
metric space and associated each fuzzy metric space to a Hausdorff topology.
Till now many topological structures and related theories have been defined and
studied on the probabilistic metric space and George and Veeramani’s fuzzy metric
space. For example, H¨ohle [7, 8] studied the associated topologies and the fuzzy unifor-
mities in the probabilistic metric space, J. Guti´errez Garc´ıa and M.A. de Prada Vicente
[6] studied the Hutton [0,1]-quasi-uniformities generated by the George and Veera-
mani’s fuzzy metric. Gregori,etc,in [4, 5] studied the convergence and completeness in
George and Veeramani’s fuzzy metric spaces. Recall that the value M(x,y,t)in the def-
inition of George and Veeramani’s fuzzy metric can be thought as the degree of the
nearness between xand ywith respect to t. Hence in this paper, we want to give the
degree convergence theory of sequence in fuzzy metric spaces, and generalize the cor-
responding theory of convergence and compactness in classical metric spaces to fuzzy
metric spaces.
67

2 Convergence in fuzzy metric spaces
Since the value M(x,y,t)can be thought as the degree of the nearness between xand y
with respect to t, in this section, we will give the definitions of degree convergence and
study the relationship between them.
Definition 1. Let (X,M)be a fuzzy metric space, x ∈X and {xn}be sequence. The
degree to which {xn}converges to x is defined by
Con({xn},x) = ^
ε>0_
N∈N
^
n>N
M(xn,x,ε).
The degree to which {xn}accumulates to x is defined by
Ad({xn},x) = ^
ε>0^
N∈N
_
n>N
M(xn,x,ε).
The degree to which {xn}is a Cauchy sequence is defined by
Cauchy({xn}) = ^
ε>0^
N∈N
_
n,m>N
M(xn,xm,ε).
Lemma 1. Let (X,M)be a fuzzy metric space, x ∈X and {xn}be sequence. Then we
have the following results:
(1) Con({x},x) = Ad({x},x) = Cauchy({x}) = 1, where {x}is the constant se-
quence of x;
(2) Con({xn},x)≤Ad({xn},x);
(3) Con({xn},x)≤Cauchy({xn})for all x ∈X ;
(4) Ad({xn},x) = W{xnk}Con({xnk},x);
(5) Ad({xn},x)≤W{xnk}Con({xnk},x).
(6) Ad({xn},x)∧Cauchy({xn})≤Con({xn},x).
Example 1. Let dbe an ordinary metric on Xand Mdbe the induced fuzzy metric.
In the following, we know that the convergence in (X,Md)is coincident with that in
(X,d).
Con({xn},x) = ^
ε>0_
N∈N
^
n>N
ε
ε+d(xn,x)=^
ε>0
ε
ε+VN∈NWn>Nd(xn,x)=1,xn→x,
0,others,
Cauchy({xn},x) = ^
ε>0_
N∈N
^
n,m>N
ε
ε+d(xn,xm)=^
ε>0
ε
ε+VN∈NWn,m>Nd(xn,xm)
=1,{xn}is Cauchy,
0,others,
Ad({xn},x) = ^
ε>0^
N∈N
_
n>N
ε
ε+d(xn,x)=^
ε>0
ε
ε+WN∈NVn>Nd(xn,x)=1,xn∞x,
0,others,
68

Example 2. Let X={x,y}and d:X×X×(0,+∞)→[0,1]be defined by
M(a,b,t) = 






1,a=b=x,
1,a=b=y,
1,a6=b,t>1
2,
1
2+t,a6=b,t≤1
2,
Then dis a fuzzy metric on Xand Con({x},y) = Ad({x},y) = 1
2. If we take {xn}=
{x,y,x,y,x...}, then Con({xn},x) = 0 and Ad({xn},x) = 1.
3 Compactness in fuzzy metric spaces
In this section, we want to generalized the compactness in metric spaces to fuzzy setting
according to the above convergence theory.
Definition 2. Let (X,M)be a fuzzy metric space. The degree to which (X,M)is com-
pact is defined by Comp(M) = ^
{xn}
_
x∈X
Ad({xn},x).
The degree to which (X,M)is sequently compact is defined by
Scomp(M) = ^
{xn}
_
{xnk}
_
x∈X
Con({xnk},x).
Definition 3. Let (X,M)be a fuzzy metric space and F ∈2X. The degree to which F is
an ε-net of (X,M)is defined by
ε−net(F) = ^
x∈X
_
y∈F
M(x,y,ε).
The degree to which (X,M)is totally bounded is defined by
Totallb(M) = ^
ε>0_
F∈2(X)
ε−net(F).
The degree to which (X,M)is complete is defined by
Complete(M) = ^
{xn}
(Cauchy({xn})→_
x∈X
Con({xn},x))
Theorem 1. Let (X,M)be a fuzzy metric space. Then Com p(M) = Scomp(M).
Theorem 2. Let (X,M)be a fuzzy metric space. Then
Totallb(M) = ^
{xn}
_
{xnk}
Cauchy({xnk}).
Theorem 3. Let (X,M)be a fuzzy metric space. Then Comp(M) = Complete(M)∧
Totallb(M).
69

Acknowledgements. This work is supported by Natural Science foundation of Shan-
dong Province (No. ZR2011AQ010).
References
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64 (1994) 395–399.
2. A. George, P. Veeramani, Some theorems in fuzzy metric spaces, J. Fuzzy Math. 3 (1995)
933–940.
3. A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets
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spaces, Topology and its Applications 156 (2009) 3002–3006.
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fuzzy (quasi-)metric spaces, Fuzzy Sets and Systems 157 (2006) 755–766.
7. U. H¨ohle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math. 38
(1982) 289–323.
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(1982) 63–69.
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Boston, 2001.
10. I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (5)
(1975) 336–344.
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125(1987) 141–153.
12. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28 (1942) 535–537.
13. S. E. Rodabaugh, Axiomatic foundations for uniform operators quasi-uniformites, pp. 199–
233, Chapter 7 in S. E. Rodabaugh, E. P. Klement, eds., Topological and Algebraic Struc-
tures in Fuzzy Sets: A Handbook of Recent Development in the Mathematics of Fuzzy Sets,
Trends in Logic 20(2003), Kluwer Academic Publishers (Boston/Dordrecht/London).
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15. Y. Yue, F. G. Shi,On fuzzy pseudo-metric spaces, Fuzzy Sets and Systems 161 (2010) 1105–
1106.
70

Dept. of K no wledge-Based Ma th em at ical System s
Fu zzy Lo gi c Laborat or iu m Li nz-Hagenber g
Jo ha nn es K ep ler Unive rs it y
A- 40 40 L in z, Austria
Te l. : +4 3 73 2 2468 41 40
Fa x: + 43 7 32 24 68 4142
E- Ma il : sabine.lumpi@jku.at
WW W : http://www.flll.jku.at/
Softwa re C om petence Cen te r Ha genberg
Softwarepark 21
A- 42 32 H agen berg, Aus tr ia
Te l. : +4 3 7236 33 43 8 00
Fa x: + 43 7 236 334 3 88 8
E- Ma il : sekr@scch.at
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