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Annals of Fuzzy Mathematics and Informatics
Volume 14, No. 3, (September 2017), pp. 315–330
ISSN: 2093–9310 (print version)
ISSN: 2287–6235 (electronic version)
http://www.afmi.or.kr
@FMI
c
Kyung Moon Sa Co.
http://www.kyungmoon.com
Fuzzy congruence relations on almost distributive
lattices
Berhanu Assaye Alaba, Gezahagne Mulat Addis
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@FMI
@FMI
@FMI
@FMI
@FMI
@FMI@FMI
@FMI@FMI
@FMI@FMI
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Reprinted from the
Annals of Fuzzy Mathematics and Informatics
Vol. 14, No. 3, September 2017
Annals of Fuzzy Mathematics and Informatics
Volume 14, No. 3, (September 2017), pp. 315–330
ISSN: 2093–9310 (print version)
ISSN: 2287–6235 (electronic version)
http://www.afmi.or.kr
@FMI
c
Kyung Moon Sa Co.
http://www.kyungmoon.com
Fuzzy congruence relations on almost distributive
lattices
Berhanu Assaye Alaba, Gezahagne Mulat Addis
Received 25 May 2017;Revised 23 June 2017;Accepted 12 July 2017
Abstract. In this paper we give several characterizations for fuzzy
ideals, fuzzy homomorphisms, and fuzzy congruences of an almost dis-
tributive lattice. In addition, the quotient of an almost distributive lattice
induced by a fuzzy congruence is also presented in the paper. Furthermore,
we obtain a kind of fuzzy congruences for which their quotient is a distribu-
tive lattice and for which it is not. Mainly, we construct a monomorphism
of the lattice of fuzzy ideals into the lattice of fuzzy congruences of almost
Boolean rings, and we give a necessary and sufficient condition for this
monomorphism to become a lattice isomorphism.
2010 AMS Classification: 06D72, 06F15, 08A72
Keywords: ADLs, Fuzzy ideals, Fuzzy homomorphisms, Fuzzy congruences.
1. Introduction
The concept of an almost distributive lattice(ADL) was first introduced by U.M.
Swamy and G. C. Rao [9] in 1980 as a common abstraction to most of the existing
ring theoretic and lattice theoretic generalization of Boolean algebras. An ADL is an
algebra with two binary operations ∨and ∧which satisfies almost all the properties
of a distributive lattice with smallest element 0 except possibly the commutativity
of ∨, the commutativity of ∧and the right distributivity of ∨over ∧. It was also
observed that any one of these three properties converts an ADL into a distributive
lattice. The study of ideals, and congruence relations on ADLs was initiated in [9]
and later studied by many authors. In most of algebraic structures the concept
of congruences is closely related with structures such as; normal subgroups (in the
case of groups), ideals (in the case of rings), and quotient algebras. This makes the
study of congruences more important both from theoretical stand point and for its
applications in many fields. In this view, the concept of filter congruences and factor
congruences was introduced in an ADL analogous to that in a distributive lattice by
Gezahagne Mulat Addis et al. /Ann. Fuzzy Math. Inform. 14 (2017), No. 3, 315–330
U.M. Swamy et al. [8]. Following this, Y. S. Pawar et al. [4] further studied on the
class of congruences on ADLs induced by multiplicatively closed sets.
On the other hand, the study of fuzzy sets was done in 1965 by L. A. Zadeh [11].
Since then many authors have been studying fuzzy subalgebras of several algebraic
structures. Rosenfeld [5] in 1971 developed the concept of fuzzy subgroup. W. J.
Liu [2] in 1982 initiated the study of fuzzy subrings, and fuzzy ideals of a ring. D.S.
Malik et al. [3] studied fuzzy homomorphisms of rings. In 1990, Y. Bo et al. [10]
introduced the concept of fuzzy ideals and fuzzy congruences of distributive lattices
and showed that if L is relatively complemented distributive lattice with zero, then
there is a one-to-one correspondence between the lattice of fuzzy ideals and the
lattice of fuzzy congruences of L. Later in 1998 U. M. Swamy et al. [7] studied
properties of L-fuzzy ideals and L-fuzzy congruences of lattices.
More recently, U. M. Swamy et al. [6] initiated the study of L-fuzzy ideals of
ADLs. They particularly proved that the class of L-fuzzy ideals of an ADL forms
a complete distributive lattice. In this paper, we extend the results in [6] and give
several characterizations for fuzzy ideals, fuzzy homomorphisms, and fuzzy congru-
ences of ADLs. Quotient ADLs induced by fuzzy congruences are also presented
in the paper. In addition, we obtain a kind of fuzzy congruences for which their
quotient is a distributive lattice and for which it is not. Furthermore, we give the
smallest fuzzy congruence on an ADL Asuch that its quotient is a distributive lat-
tice. Finally, we construct a monomorphism of the lattice of fuzzy ideals and the
lattice of fuzzy congruences of almost Boolean rings and we give a necessary and
sufficient condition for this monomorphism to become an order isomorphism.
Most of the results in the paper seem analogous to those results in distributive
lattices, though the proofs are different due to the absence of the commutativity of
∨and ∧.
2. Preliminaries
In this section we recall some definitions and basic results on almost distributive
lattices.
Definition 2.1 ([9]).An algebra (A, ∨,∧,0) of type (2,2,0) is called an almost
distributive lattice, abbreviated as ADL, if it satisfies the following axioms:
(i) a∨0 = a,
(ii) 0 ∧a= 0,
(iii) (a∨b)∧c= (a∧c)∨(b∧c),
(iv) a∧(b∨c)=(a∧b)∨(a∧c),
(v) a∨(b∧c)=(a∨b)∧(a∨c),
(v) (a∨b)∧b=b, for all a, b, c ∈A.
Lemma 2.2 ([9]).For any a∈A, we have
(1) a∧0=0,
(2) 0 ∨a=a,
(3) a∧a=a,
(4) a∨a=a.
Lemma 2.3 ([9]).For any a, b ∈A, we have
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(1) (a∧b)∨b=b,
(2) a∨(a∧b) = a=a∧(a∨b),
(3) a∨(b∧a) = a= (a∨b)∧a,
(4) ∧is associative and a∧b∧c=b∧a∧c.
Corollary 2.4 ([9]).For any a, b ∈A, we have
(1) a∨b=aif and only if a∧b=b,
(2) a∨b=bif and only if a∧b=a.
Definition 2.5 ([9]).For any a, b ∈A, we say that ais less than or equals to band
we write a≤b, if a∧b=aor equivalently a∨b=b.
Theorem 2.6 ([9]).For any a, b ∈A, the following are equivalent:
(1) (a∧b)∨a=a,
(2) a∧(b∨a) = a,
(3) (b∧a)∨b=b,
(4) b∧(a∨b) = b,
(5) a∧b=b∧a,
(6) a∨b=b∨a,
(7) the supremum of aand bexists in Aand equals to a∨b,
(8) there exists x∈Asuch that a≤xand b≤x,
(9) the infimum of aand bexists in Aand equals to a∧b.
Definition 2.7 ([9]).A nonempty subset Iof an ADL Ais called an ideal of A, if
a∨b, a ∧x∈I , for all a, b ∈Iand for all x∈A.
It can be observed that x∧a∈Ifor all a∈Iand all x∈A. For any subset
S⊆A, the smallest ideal of Acontaining Sis called the ideal of Agenerated by S
and is denoted by hS]. Note that:
hS] = {(Wxi)∧a:a∈A, xi∈S, i = 1, ..., n for some n∈Z+}.
If S={a},then we write ha],for hS]. In this case, ha] = {a∧x:x∈A}.
3. Fuzzy ideals and Fuzzy homomorphisms on ADLs
In this section, we give several characterizations for fuzzy ideals and fuzzy homo-
morphisms of ADLs. Some of the results on fuzzy ideals are due to [6]. Remember
that, for any set A, a function µ:A→[0,1] is called a fuzzy subset of A. For each
t∈[0,1] ,the set
µt={x∈A:µ(x)≥t}
is called the level subset of µat t[11].
Definition 3.1. A fuzzy subset µof Ais called a fuzzy subADL of A, if:
µ(x∨y)∧µ(x∧y)≥µ(x)∧µ(y),for all x, y ∈A.
Definition 3.2 ([6]).A fuzzy subset µof A is called a fuzzy ideal of A, if:
µ(0) = 1 and µ(x∨y) = µ(x)∧µ(y),for all x, y ∈A.
We denote the class of all fuzzy ideals of Aby F I(A).
Example 3.3. Let A={0, a, b, c}and let ∨and ∧be binary operations on A
defined by:
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∨0a b c
0 0 a b c
a a a a a
b b b b b
c c a b c
∧0a b c
0 0 0 0 0
a0a b c
b0a b c
c0c c c
Then (A, ∨,∧,0) is an ADL(a discrete ADL) [9]. Now define a fuzzy subset µof A
by:
µ(0) = 1, µ(a)=0.6 = µ(b) and µ(c)=0.8.
Thus µis a fuzzy ideal of A.
Lemma 3.4. [6]A fuzzy subset µof Ais a fuzzy ideal of Aif and only if
(1) µ(0) = 1,
(2) µ(x∨y)≥µ(x)∧µ(y),
(3) µ(x∧y)≥µ(x)∨µ(y), for all x, y ∈A,
Lemma 3.5. Let µbe fuzzy subADL of A. Then µis a fuzzy ideal of Aif and only
if
µ(0) = 1 and a∧b=b⇒µ(a)≤µ(b),for all a, b ∈A
Lemma 3.6 ([6]).Let µbe fuzzy subset of A. Then µis a fuzzy ideal of Aif and
only if µtis an ideal of A, for all t∈[0,1] .
Lemma 3.7. The intersection of any family of fuzzy ideals of Ais a fuzzy ideal.
Remark 3.8. Note that the union of a family of fuzzy ideals of Ais not in general
a fuzzy ideal of A. We verify this in the following example:
Example 3.9. Let A={0, a, b, c}and let ∨and ∧be binary operations on Ade-
fined by:
∨0a b c
0 0 a b c
a a a a a
b b a b a
c c a a c
∧0a b c
0 0 0 0 0
a0a b c
b0b b 0
c0c0c
Then it is clear that (A, ∨,∧,0) is an ADL. Now define fuzzy subsets µand σof A
by:
µ(0) = 1, µ(a)=0.5 = µ(b) and µ(c)=0.7,
σ(0) = 1, σ(a)=0.6 = σ(c) and σ(b)=0.8.
Thus both µand σare fuzzy ideals of Abut µ∪σfails to be a fuzzy ideal of A.
Lemma 3.10 ([6]).A nonempty subset Iof Ais an ideal of Aif and only if the
characteristic function χIof Iis a fuzzy ideal of A.
Definition 3.11. Let µbe fuzzy subset of A. The smallest fuzzy ideal of Acon-
taining µis called a fuzzy ideal of Ainduced by µand is denoted by hµ].
Lemma 3.12. For any fuzzy subset µof A,
hµ] = T{σ∈F I (A) : µ⊆σ}.
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Lemma 3.13. Let Sbe any subset of Aand χSits characteristic function. Then
hχS] = χhS].
Proof. To prove that hχS] = χ(S],we show that χhS]is the smallest fuzzy ideal of A
containing χS. Since hS] is an ideal of Acontaining S, it is clear that χhS]is a fuzzy
ideal of A containing χS. It remains to show that it is the smallest fuzzy ideal of A
containing χS. Let µbe any fuzzy ideal of Acontaining χS, that is, χS(x)≤µ(x),
for all x∈S, then µ(x) = 1,for all x∈S.
Now consider y∈ hS]. Then y= (Wxi)∧a, for some a∈A, xi∈S, i = 1, ..., n;n∈
Z+.Then, for each y∈ hS],we have
µ(y) = µ((_xi)∧a)≥µ(_xi)∨µ(a)≥µ(_xi)≥^µ(xi)=1.
Thus χhS](y)≤µ(y),for all y∈A. So χhS]⊆µ. Hence the result holds.
For any fuzzy subset µof A, it is clear that
µ(x) = Sup{α∈[0,1] : x∈µα},for all x∈A.
In the following theorem, we characterize a fuzzy ideal induced by fuzzy sets.
Theorem 3.14. Let µbe a fuzzy subset of A. Then a fuzzy subset bµof Adefined
by:
bµ(x) = Sup{α∈[0,1] : x∈ hµα]},for all x∈A
is a fuzzy ideal of A induced by µ.
Proof. It is enough if we show that bµis the smallest fuzzy ideal of Acontaining µ.
Clearly bµis a fuzzy subset of A. Also bµ(0) = S up{α∈[0,1] : 0 ∈ hµα]}. Since hµα]
is an ideal of A, for all α∈[0,1], 0 ∈ hµα],for all α∈[0,1]. Then it follows that
bµ(0) = Sup{α∈[0,1]}= 1.
Next we show that bµ(x∨y)≥bµ(x)∧bµ(y),for all x, y ∈A. For;
bµ(x)∧bµ(y) = Sup{α∈[0,1] : x∈ hµα]} ∧ Sup{β∈[0,1] : y∈ hµβ]}
=Sup{min{α, β }:x∈ hµα], y ∈ hµβ]}.
If we put λ=min{α, β}, then λ≤αand λ≤β, which implies that hµα]⊆ hµλ] and
hµβ]⊆ hµλ]. That is, if x∈ hµα] and y∈ hµβ], then x, y ∈ hµλ].Thus x∨y∈ hµλ].
So
bµ(x)∧bµ(y) = Sup{min{α, β }:x∈ hµα], y ∈ hµβ]}
≤Sup{λ∈[0,1] : x∨y∈ hµλ]}
=bµ(x∨y).
Next we show that bµ(x∧y)≥bµ(x)∨bµ(y). It follows from the definition of
bµthat bµ(b)≥bµ(a),whenever a∧b=b, for all a, b ∈A. Using this fact, Since
x∧(x∧y) = x∧yand y∧(x∧y) = x∧y, for all x, y ∈A, we get that bµ(x∧y)≥bµ(x)
and bµ(x∧y)≥bµ(y) which implies that bµ(x∧y)≥bµ(x)∨bµ(y). Then bµis a fuzzy
ideal of A.
Next we show that µ⊆bµ. For any x∈A, put µ(x) = λ. Then x∈µλ⊆ hµλ].
Thus x∈ hµλ].So λ∈ {α∈[0,1] : x∈ hµα]}, that is,
bµ(x) = Sup{α∈[0,1] : x∈ hµα]} ≥ λ=µ(x).
Hence µ⊆bµ.
Now it remains to show that bµis the smallest fuzzy ideal such that µ⊆bµ. Let
γbe any fuzzy ideal of Asuch that µ⊆γ. Then µα⊆γα,for all α∈[0,1]. For;
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x∈µα⇒µ(x)≥α⇒γ(x)≥α⇒x∈γα. Since γis a fuzzy ideal of A, we have γα
is an ideal of A, for all α∈[0,1]. That is, γαis an ideal of Acontaining µα. Thus
hµα]⊆γα.
Now for any x∈A, consider
bµ(x) = Sup{α∈[0,1] : x∈ hµα]} ≤ Sup{α∈[0,1] : x∈γα}=γ(x).
Hence the result holds.
Theorem 3.15 ([6]).The class F I (A)of all fuzzy ideals of Aforms a complete
lattice where the infimum and supremum of any family {µα:α∈∆}of fuzzy ideals
is given by: Vµα=∩µαand Wµα=h∪µα].
In the remaining part of this section we define fuzzy homomorphisms on ADLs
and we present some results on fuzzy homomorphisms in connection with fuzzy
ideals.
Recall from [1] that, for any sets Aand Ba mapping f:A×B→[0,1] is
called a fuzzy relation of Ainto B. A fuzzy relation fof Ainto Bis called a
fuzzy mapping if for each x∈Athere exists a unique element yx∈Bsuch that
f(x, yx) = 1 in this case we call this unique element yxa fuzzy image of xunder f.
We write f:A99K B, for a fuzzy mapping fof Ainto B. Image of fis the set
{yx:x∈A}={y∈B:f(x, y)=1}. Moreover, for any y∈B,
f−1(y) = {x∈A:yx=y}={x∈A:f(x, y)=1}.
As usual, fis said to be onto, if for each y∈B, there exists x∈Asuch that yx=y
and fis said to be one-one, if for each a, b ∈A,ya=yb=⇒a=b.
Definition 3.16. Let Aand Bbe ADLs. A fuzzy mapping f:A99K Bis called a
fuzzy homomorphism of ADLs, if the following are satisfied, for all a, b ∈A:
(i) y0= 0 (a zero element in B),
(ii) f(x1∨x2, y)≥sup{f(x1, y1)∧f(x2, y2) : y=y1∨y2, y1, y2∈B},
(iii) f(x1∧x2, y)≥sup{f(x1, y1)∧f(x2, y2) : y=y1∧y2, y1, y2∈B}.
Lemma 3.17. Let f:A99K Bbe a fuzzy homomorphism of ADLs. Then we have
the following:
(1) y(a∨b)=ya∨yb,
(2) y(a∧b)=ya∧yb,
for all a, b ∈A.
Proof. We have yaand ybare the unique elements in Bsuch that f(a, ya) = 1 and
f(b, yb) = 1. We show that f(a∨b, ya∨yb) = 1. Put z=ya∨ybfor simplicity. Then
f(a∨b, z) = Sup{f(a, z1)∧f(b, z2) : z=z1∨z2, z1, z2∈B}
≥f(a, ya)∧f(b, yb)
= 1.
Since y(a∨b)is the unique element in Bsuch that f(a∨b, y(a∨b)) = 1, we get that
y(a∨b)=ya∨yb. Similarly, it can be verified that y(a∧b)=ya∧yb.
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Lemma 3.18. Let Aand Bbe ADLs and fa fuzzy homomorphism of Aonto
B. Let µbe a fuzzy subADL(respectively a fuzzy ideal) of Aand σbe a fuzzy sub-
ADL(respectively a fuzzy ideal) of B. Then
(1) f(µ)is a fuzzy subADL(respectively a fuzzy ideal) of B,
(2) f−1(σ)is a fuzzy subADL(respectively a fuzzy ideal) of A.
Theorem 3.19. Let Aand Bbe ADLs and f:A→Ba mapping. Then fis a
homomorphism if and only if its characteristic mapping χfis a fuzzy homomorphism
of Ainto B, where χf:A×B→[0,1] is defined as:
χf(a, b) = (1if f(a) = b
0otherwise,
for all (a, b)∈A×B.
Theorem 3.20. Let f be a fuzzy homomorphism of Ainto B. Then a subset f∗of
Adefined by
f∗={x∈A:f(x, 0) = 1}
is an ideal of A.
Proof. Clearly, f(0,0) = 1. then 0 ∈f∗. Let a, b ∈f∗. Then f(a, 0)=1=f(b, 0).
We show that f(a∨b, 0) = 1, for;
f(a∨b, 0) = sup{f(a, y1)∧f(b, y2) : 0 = y1∨y2and y1, y2∈B}
≥f(a, 0) ∧f(b, 0) = 1.
That is, f(a∨b, 0) = 1. Thus a∨b∈f∗. Also let a∈f∗and x∈A. Then
f(a, 0) = 1. Now consider
f(a∧x, 0) = Sup{f(a, y1)∧f(x, y2) : 0 = y1∧y2} ≥ f(a, 0) ∧f(x, yx)=1,
that is, f(a∧x, 0) = 1.Thus a∧x∈f∗. So f∗is an ideal of A.
Theorem 3.21. Let fbe a fuzzy homomorphism of Ainto B. Then a fuzzy subset
µfof Adefined by
µf(x) = f(x, 0),for all x∈A
is a fuzzy ideal of A.
Proof. Clearly µf(0) = 1. For any a, b ∈A, consider the following:
µf(a∨b) = f(a∨b, 0)
=sup{f(a, y1)∧f(b, y2) : y1∨y2= 0 and y1, y2∈B}
≥f(a, y1)∧f(b, y2)∀y1, y2∈B, with y1∨y2= 0.
In particular, for y1= 0 and y2= 0. That is, µf(a∨b)≥f(a, 0) ∧f(b, 0) =
µf(a)∧µf(b).
Also,
µf(a∧b) = f(a∧b, 0)
=sup{f(a, y1)∧f(b, y2) : 0 = y1∧y2andy1, y2∈B}
≥f(a, y1)∧f(b, y2)∀y1, y2∈B, with y1∧y2= 0.
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In particular, for y1= 0 and y2=yb. That is, µf(a∧b)≥f(a, 0) ∧f(b, yb) =
f(a, 0) ∧1 = f(a, 0) = µf(a). Similarly, doing we get µf(a∧b)≥µf(b) which
implies that µf(a∧b)≥µf(a)∨µf(b). Thus µfis a fuzzy ideal of A.
4. Fuzzy congruences on ADLs
In this section we define fuzzy congruence relations on ADLs and we give several
characterizations for fuzzy congruences in terms of fuzzy ideals and fuzzy homomor-
phisms.
Recall that for any set Aa fuzzy subset Θ of A×Ais called a fuzzy relation on A.
Definition 4.1. A fuzzy relation Θ on an ADL Ais a called fuzzy congruence
relation on A, if the following are satisfied:
(i) Θ(a, a)=1,for all a∈A,
(ii) Θ(a, b) = Θ(b, a),for all a, b ∈A,
(iii) Θ(a, c)≥Θ(a, b)∧Θ(b, c),for all a, b, c ∈A,
(iv) Θ(a∨c, b ∨d)∧Θ(a∧c, b ∧d)≥Θ(a, b)∧Θ(c, d),for all a, b, c, d ∈A.
We denote the set of all fuzzy congruence relations on Aby FC(A).
Example 4.2. Let Abe an ADL as in Example 3.3. Define a fuzzy relation Θ on
Aas follows:
Θ(0,0) = Θ(a, a) = Θ(b, b) = Θ(c, c) = 1,
Θ(0, c) = Θ(c, 0) = 0.8,
Θ(a, b) = Θ(b, a) = Θ(a, c) = Θ(c, a) = Θ(b, c) = Θ(c, b)
= Θ(0, a) = Θ(a, 0) = Θ(0, b) = Θ(b, 0) = 0.7.
Then Θ is a fuzzy congruence relation on A.
Lemma 4.3. Let θbe an equivalence relation on A. Then θis a congruence relation
on Aif and only if its characteristic function χθis a fuzzy congruence on A.
Lemma 4.4. A fuzzy relation Θon Ais a fuzzy congruence on Aif and only if
every level subset Θtof Θat t∈[0,1] is a congruence relation on A.
Theorem 4.5. Let Θbe a fuzzy congruence relation on A. A fuzzy subset µΘdefined
by µΘ(x) = Θ(x, 0) for all x∈Ais a fuzzy ideal of A.
Proof. The proof is analogous to that of Theorem 3.21.
Theorem 4.6. Let Θbe a fuzzy congruence relation on A. A fuzzy subset νΘdefined
by νΘ(x) = Inf{Θ(a∧x, x) : a∈A},for all x∈Ais a fuzzy ideal of A.
Proof. νΘ(0) = Inf {Θ(a∧0,0) : a∈A}=Inf{Θ(0,0) : a∈A}= Θ(0,0) = 1. For
any x, y ∈A, consider
νΘ(x∨y) = Inf{Θ(a∧(x∨y), x ∨y) : a∈A}
=Inf{Θ[(a∧x)∨(a∧y), x ∨y] : a∈A}
≥Inf{Θ(a∧x, x)∧Θ(a∧y, y) : a∈A}
=Inf{Θ(a∧x, x) : a∈A} ∧ Inf {Θ(a∧y, y ) : a∈A}
=νΘ(x)∧νΘ(y).
Also consider
νΘ(x∧y) = Inf{Θ(a∧(x∧y), x ∧y) : a∈A}
=Inf{Θ[(a∧x)∧y], x ∧y) : a∈A}
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≥Inf{Θ(a∧x, x)∧Θ(y, y) : a∈A}
=Inf{Θ(a∧x, x) : a∈A}
=νΘ(x).
In the similar fashion, we get νΘ(x∧y)≥νΘ(y). Then νΘ(x∧y)≥νΘ(x)∨νΘ(y).
Thus νΘis a fuzzy ideal of A.
Theorem 4.7. Let Θbe a fuzzy congruence relation on A. Then µΘ=νΘ.
Proof. For any fuzzy congruence relation on A, we claim to show that µΘ=νΘ. For
any x∈A, we have νΘ(x) = Inf {Θ(a∧x, x) : a∈A}. Then νΘ(x)≤Θ(a∧x, x),
for all a∈A. In particular, for a= 0,
νΘ(x)≤Θ(0 ∧x, x) = Θ(0, x) = Θ(x, 0) = µΘ(x).
On the other hand, for any a∈A, consider
Θ(a∧x, x) = Θ[a∧x, (a∧x)∨x]
≥Θ(a∧x, a ∧x)∧Θ(0, x)
= Θ(x, 0) = µΘ(x).
Thus Θ(a∧x, x)≥µΘ(x),for all a∈A. So
νΘ(x) = Inf{Θ(a∧x, x) : a∈A} ≥ µΘ(x).
Hence µΘ=νΘ.
Theorem 4.8. Let f:A99K Bbe a fuzzy homomorphism. Define a fuzzy kernel of
fdenoted by Kf:A×A:→[0,1] as follows:
Kf(a, b) = (1if ya=yb
0otherwise,
for all a, b ∈A. Then this Kfis a fuzzy congruence relation on A.
Corollary 4.9. fis a fuzzy monomorphism if and only if its kernel Kfis the
characteristic function of the diagonal of A.
Theorem 4.10. Let Θbe a fuzzy congruence relation on A. For any x∈A, define
a subset Θxof Aby
Θx={y∈A: Θ(x, y)=1}.
Then Θ0is an ideal of A.
Remark 4.11. This Θ0is a level subset of a fuzzy ideal µΘ(given in Theorem 4.5)
at t= 1. Let Θ be a fuzzy congruence on A. For any x∈Aconsider a subset Θxof
Agiven by Θx={y∈A: Θ(x, y)=1}. Then we have the following properties:
(1) For any x, y ∈Aeither Θx∩Θy=∅or Θx= Θy,
(2) x∈Θyif and only if Θx= Θyor equivalently if Θ(x, y)=1.
Put A
Θ={Θx:x∈A}and define operations ∧and ∨on A
Θas follows:
Θx∧Θy= Θx∧yand Θx∨Θy= Θx∨y(*)
Then ( A
Θ,∧,∨,Θ0) becomes an ADL with Θ0as its zero element and it is called a
quotient ADL induced by the fuzzy congruence Θ on A.
Definition 4.12. A fuzzy subset µof Ais said to be multiplicatively closed, if
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µ(x∧y)≥µ(x)∧µ(y),for all x, y ∈A.
Let λbe a multiplicatively closed fuzzy subset of Awith Sup{λ(x) : x∈A}= 1.
Define fuzzy relations Ψλand Φλon Ainduced by λas follows:
Ψλ(x, y) = Sup{λ(a) : x∧a=y∧a, a ∈A}and
Φλ(x, y) = Sup{λ(b) : b∧x=b∧y, b ∈A}.for all x, y ∈A.
Then we have the following results.
Theorem 4.13. Ψλis a fuzzy congruence relation on Aand the quotient A
Ψλis
a distributive lattice. Moreover if Ahas maximal elements then the quotient A
Ψλ
becomes bounded with the class of all maximal elements, its unit element and {0}its
least element.
Proof. We first show that Ψλis a fuzzy congruence on A. For; for any x, y, z ∈A,
consider the following:
(1) Ψλ(x, x) = Sup{λ(a) : x∧a=x∧a, a ∈A}=Sup{λ(a) : a∈A}= 1,
(2) Ψλ(x, y) = Sup{λ(a) : x∧a=y∧a, a ∈A}= Ψλ(y, x),
(3) for any a, b ∈A, if x∧a=y∧aand y∧b=z∧b, then we get x∧(a∧b) = y∧(a∧b)
and y∧(a∧b) = z∧(a∧b) which implies that x∧(a∧b) = z∧(a∧b).
Now consider
Ψλ(x, y)∧Ψλ(y, z )
=Sup{λ(a) : x∧a=y∧a, a ∈A} ∧ Sup{λ(b) : y∧b=z∧b, b ∈A}
=Sup{λ(a)∧λ(b) : x∧a=y∧a and y ∧b=z∧b, a, b ∈A}
≤Sup{λ(a∧b) : x∧a=y∧a and y ∧b=z∧b, a, b ∈A}
≤Sup{λ(c) : x∧c=y∧c, c ∈A}
= Ψλ(x, z).
(4) similarly, it can be verified that
Ψλ(x1∨x2, y1∨y2)≥Ψλ(x1, y1)∧Ψλ(x2, y2)
and
Ψλ(x1∧x2, y1∧y2)≥Ψλ(x1, y1)∧Ψλ(x2, y2).
Thus Ψλis a fuzzy congruence relation on A.
Next we show that the quotient A
Ψλis a distributive lattice. Clearly, it is an ADL
together with binary operations ∨and ∧defined as in (*). It suffices to show that
either ∧or ∨is commutative on A
Ψλ. For any x, y ∈A, consider
Ψλ(x∧y, y ∧x) = sup{λ(a):(x∧y)∧a= (y∧x)∧a, a ∈A}=sup{λ(a) : a∈A}= 1.
Then Ψλ
x∧y= Ψλ
y∧xwhich says that ∧is commutative. Thus the quotient A
Ψλis a
distributive lattice.
Remark 4.14. It is clear that Φλis a fuzzy congruence on A. But the quotient
A
Φλis not in general a distributive lattice. We verify this by giving the following
example.
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Example 4.15. Let Abe a discrete ADL with |A| ≥ 3 (see [9] and [4]). Let λbe a
fuzzy subset of Adefined by:
λ(x) = (0if x= 0
1otherwise.
Then λis a multiplicatively closed fuzzy subset of Aand Φλis a fuzzy congruence
on A. We show that A
Φλ∼
=A. For; consider the canonical map f:A→A
Φλdefined
by:
f(x)=Φλ
x,for all x∈A.
Then it is clear that fis an epimorphism. It remains to show that fis one-one. For
any x, y ∈A
f(x) = f(y)⇒Φλ
x= Φλ
y
⇒Φλ(x, y)=1
⇒Sup{λ(a) : a∧x=a∧y}= 1.
Since Imgλ ={0,1}, there exists a nonzero a∈Asuch that a∧x=a∧y. By
the fact that every nonzero element in Ais maximal, it follows that x=y. Thus
fis one-one and hence an isomorphism. Since Ais a discrete ADL with at least 3
elements, it is not a lattice. So the quotient A
Φλis not a distributive lattice.
In the next theorem, we give the smallest fuzzy congruence on Afor which its
quotient is a distributive lattice.
Theorem 4.16. A fuzzy relation ηon Adefined by
η(a, b) = (1if ha] = hb]
0otherwise,
for all a, b ∈A, is a fuzzy congruence relation on Aand it is the smallest such that
the quotient A
ηis a distributive lattice.
Proof. Clearly, ηis a fuzzy congruence on A. It is also clear that ha∧b] = hb∧a],
for all a, b ∈A. Then η(a∧b, b ∧a) = 1. Thus η(a∧b)=η(b∧a)which implies that the
quotient A
ηis a distributive lattice. Now let Θ be any fuzzy congruence on Asuch
that the quotient A
Θis a distributive lattice. We claim to show that η⊆Θ. For any
a, b ∈A, consider the following.
If ha]6=hb], then η(a, b)=0≤Θ(a, b). Otherwise,
ha] = hb]⇒a∧b=b, b ∧a=a
⇒Θ(a∧b)= Θb,Θ(b∧a)= Θa
⇒Θa= Θb(since A
Θis a lattice)
⇒Θ(a, b) = 1 = η(a, b).
Thus η⊆Θ.
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Definition 4.17. An ADL Ais said to be associative, if the binary operation ∨in
Ais associative.
Theorem 4.18. Let Abe an associative ADL and µa fuzzy ideal of A. Let us
define fuzzy relation φµon Aby:
φµ(x, y) = Sup{µ(a) : a∨x=a∨y, a ∈A},for all x, y ∈A.
Then φµis a fuzzy congruence relation on A.
Proof. For any x, y, z ∈A, consider
(1) φµ(x, x) = Sup{µ(a) : a∨x=a∨x, a ∈A}=Sup{µ(a) : a∈A}= 1,
(2) φµ(x, y) = Sup{µ(a) : a∨x=a∨y, a ∈A}=φµ(y, x),
(3) if a∨x=a∨yand b∨y=b∨z, for a, b ∈A, then as Ais an associative
ADL, we get (a∨b)∨x= (a∨b)∨yand (a∨b)∨y= (a∨b)∨zwhich implies that
(a∨b)∨x= (a∨b)∨z.
Now consider
φµ(x, y)∧φµ(y, z )
=Sup{µ(a) : a∨x=a∨y , a ∈A} ∧ Sup{µ(b) : b∨y=b∨z, b ∈A}
=Sup{µ(a)∧µ(b) : a∨x=a∨yand b∨y=b∨z,a, b ∈A}
=Sup{µ(a)∧µ(b) : (a∨b)∨x= (a∨b)∨z,a, b ∈A}
≤Sup{µ(a∨b):(a∨b)∨x= (a∨b)∨z,a, b ∈A}
≤Sup{µ(c) : c∨x=c∨z,c∈A}
=φµ(x, z).
(4) Similar to (3), we can verify that
φµ(x1∨x2, y1∨y2)≥φµ(x1, y1)∧φµ(x2, y2)
and
φµ(x1∧x2, y1∧y2)≥φµ(x1, y1)∧φµ(x2, y2).
Thus φµis a fuzzy congruence relation on A.
Theorem 4.19. φµis the smallest fuzzy congruence on Acontaining the product
fuzzy ideal µ×µ, of A×A, where the product of any two fuzzy subsets µand νof
Aand Brespectively is defined as:
(µ×ν)(x, y) = µ(x)∧ν(y),for all (x, y)∈A×B.
Proof. We see in the above theorem that φµis a fuzzy congruence on A. We first
show that µ×µ⊆φµ. For; for any x, y ∈A, we have (µ×µ)(x, y) = µ(x)∧µ(y) =
µ(x∨y). Put B={µ(a) : a∨x=a∨y,a∈A}. Since (x∨y)∨x=x∨y= (x∨y)∨y,
µ(x∨y)∈B. Then µ(x∨y)≤Sup B =φµ(x, y). That is, µ×µ⊆φµ. Let Γ be any
fuzzy congruence on Asuch that µ×µ⊆Γ. For any x, y ∈A, let a∈Asuch that
a∨x=a∨y. Since Γ is a fuzzy congruence on A, we have Γ(x, y)≥Γ(x, z)∧Γ(z, y ),
for all z∈A. In particular, for z=a∨x=a∨y, Γ(x, y)≥Γ(x, a ∨x)∧Γ(a∨y, y).
But
Γ(x, a ∨x) = Γ((a∧x)∨x, a ∨x)
≥Γ(a∧x, a)∧Γ(x, x)
= Γ(a∧x, a)
≥(µ×µ)(a∧x, a)
=µ(a∧x)∧µ(a)
≥µ(a).
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Similarly, we have Γ(a∨y, y)≥µ(a).Then Γ(x, y)≥µ(a),for all a∈Awith
µ(a)∈B. Thus Sup B≤Γ(x, y). So φµ(x, y)≤Γ(x, y),for all x, y ∈A. Hence the
result holds.
5. Fuzzy Ideals and fuzzy congruenes in almost Boolean rings
Definition 5.1 ([9]).An algebra (A, +,·,0) is called an almost Boolean ring abbre-
viated as ABR, if for any a, b, c, d ∈R, it satisfies the following:
(i) a+ 0 = a,
(ii) a+a= 0,
(iii) (ab)c=a(bc),
(iv) a(b+c) = ab +ac,
(v) (a+b)c=ac +bc,
(vi) {a+ (b+c)}d={(a+b) + c}d.
Definition 5.2 ([9]).An ADL (A, ∨,∧,0) is said to be relatively complemented, if
every interval is a Boolean algebra.
Lemma 5.3 ([9]).An ADL Ais relatively complemented if and only if for any
a, b ∈A, there exists x∈Asuch that a∨b=a∨xand a∧x= 0. In this case, xis
unique which we denote by ab.
Theorem 5.4 ([9]).Let (A, ∨,∧,0) be a relatively complemented ADL. Define bi-
nary operations ”·” and ”+” on Aby a·b=a∧band a+b=ab∨ba. Then (A, +,·,0)
is an almost Boolean ring. Furthermore, a∧b=a·band a∨b=a+ (b+a·b).
Theorem 5.5 ([9]).Let (A, +,·,0) be an almost Boolean ring. Define binary oper-
ations ∧and ∨by a∧b=a·band a∨b=a+ (b+a·b). Then (A, ∨,∧,0) is a
relatively complemented ADL. Furthermore, we get a·b=a∧band a+b=ab∨ba.
The above two theorems give us a duality between the class of relatively comple-
mented ADLs and the class of almost Boolean rings analogous to the well known
Stone’s duality between the class of relatively complemented lattices with 0 and the
class of Boolean Rings.
Definition 5.6. Let Abe an almost Boolean ring. A fuzzy subset µof Ais called
a fuzzy ideal of A, if the following are satisfied:
µ(0) = 1, µ(a+b)≥µ(a)∧µ(b) and µ(a·b)≥µ(a)∨µ(b),for all a, b ∈A.
Moreover, a fuzzy relation Θ on an almost Boolean ring Ais said to be a fuzzy
congruence relation on A, if
Θ(a+c, b +d)∧Θ(a·c, b ·d)≥Θ(a, b)∧Θ(c, d),for all a, b, c, d ∈A.
As a result of the duality in [9] between the class of relatively complemented ADLs
and the class of almost Boolean rings, one can easily verify that a fuzzy subset µof
Ais a fuzzy ideal of Aas an ADL (i.e., considering Aas a relatively complemented
ADL) if and only if it is a fuzzy ideal of Aas an almost Boolean ring (i.e., considering
Aas an almost Boolean ring). Similarly, a fuzzy equivalence relation θon Ais a
fuzzy congruence on Aas an ADL if and only if it is a fuzzy congruence on Aas an
almost Boolean ring.
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Theorem 5.7. Let Abe an almost Boolean ring and µbe a fuzzy ideal of A. Then
a fuzzy relation Θµdefined by:
Θµ(a, b) = µ(a+b), for all a, b ∈A
is a fuzzy congruence relation on A.
Proof. For any a, b, c ∈A, consider the following.
(1) Θµ(a, a) = µ(a+a) = µ(0) = 1.
(2) Θµ(a, b) = µ(a+b) = µ(b+a)=Θµ(b, a).
(3) For any a, b, c ∈A, let us first see that a+c= ((a+b) + (b+c))(a+c). For;
a+c= (a+c)(a+c) = (a+b+b+c)(a+c) = ((a+b)+(b+c))(a+c).
Then
Θµ(a, c) = µ(a+c) = µ{((a+b)+(b+c))(a+c)}
≥µ((a+b)+(b+c)) ∨µ(a+c)
≥µ((a+b)+(b+c))
≥µ(a+b)∧µ(b+c)
= Θµ(a, b)∧Θµ(b, c).
(4) Using similar techniques as in (3), one can verify that
Θµ(a∨c, b ∨d)≥Θµ(a, b)∧Θµ(c, d)
and
Θµ(a∧c, b ∧d)≥Θµ(a, b)∧Θµ(c, d).
Then Θµis a fuzzy congruence relation on A.
Now let us denote Θµby C(µ) to say that it is induced by the fuzzy ideal µ. On
the other hand, for any given fuzzy congruence Θ on an almost Boolean ring A, we
can define a fuzzy ideal µΘon A by µΘ(x) = Θ(x, 0),for all x∈A(see Theorem
4.5). Let us denote µΘby I(Θ) to say that it is induced by Θ. Then we have the
following results.
Lemma 5.8. Let Abe an almost Boolean ring. If µis any fuzzy ideal of A, then
I(C(µ)) = µ.
Proof. For any x∈A, consider I(C(µ))(x) = C(µ)(x, 0) = µ(x+ 0) = µ(x).Then
I(C(µ)) = µ.
Theorem 5.9. There is a monomorphism of the lattice F I(A)of all fuzzy ideals of
an almost Boolean ring Ainto the lattice F C(A)of all fuzzy congruences on A.
Proof. Consider a mapping µ7→ C(µ) of F I(A) into F C (A). It follows from the
above lemma that this mapping is a lattice monomorphism.
Theorem 5.10. The monomorphism µ7→ C(µ)of the lattice F I (A)into the lattice
F C(A)is an isomorphism if and only if Ais a generalized Boolean algebra (or simply
a Boolean Ring).
Proof. It is observed in [10] that if Ais a generalized Boolean algebra, then the
mapping µ7→ C(µ) is a lattice isomorphism.
Conversely, suppose that the mapping µ7→ C(µ) is a lattice isomorphism of
F I (A) into F C(A). We claim to show that Ais a distributive lattice. Now it
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suffices to show that the binary operation ”·” is commutative on A. Let Θ and Φ
be fuzzy relations on Adefined by:
Θ(x, y) = (1 if x=y
0 otherwise
and
Φ(x, y) = (1 if hx] = hy]
0 otherwise,
for all x, y ∈A. Then it can be easily verified that both Θ,Φ∈F C(A). Since
the mapping µ7→ C(µ) is an isomorphism, there exists µ, ν ∈F I (A) such that
C(µ) = Θ and C(ν) = Φ which will give us that both µand νare the characteristic
function of {0}. That is, µ=νwhich implies that Θ = Φ. Thus hx] = hy] if and only
if x=y, for all x, y ∈A. It follows from the fact hx·y] = hy·x] that x·y=y·x, for
all x, y ∈A. This says that Ais a generalized Boolean algebra (or simply a Boolean
ring).
Acknowledgements. We express our heart-felt thanks to referees for their ex-
tremely valuable comments and suggestions. We are also thankful to Dr. Tilahun
Bejitual for his valuable comments.
The second author wishes to acknowledge Mr. Getachew Addis for his financial
support.
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Berhanu Assaye Alaba (berhanu−assaye@yahoo.com)
Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia
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Gezahagne Mulat Addis (buttu412@yahoo.com)
Department of Mathematics, University of Gondar, Gondar, Ethiopia
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