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Algebraic Hyperstructures: Applications to
Arithmetic Functions
Madeleine Al-Tahan
madeline.tahan@liu.edu.lb
Bannari Amman Institute of Technology
International Faculty Developement Programme on
“Avante-garde Trendas in Mathematics"
BIT, June 23rd , 2020
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 1 / 54
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Abstract
In this talk, we firstly review some basic concepts about
hyperstructure theory including hypergroups, hyperrings,
and Hv-structures. Also, we discuss some fundamental
equivalence relations on the mentioned (weak)hyper-
structures. Next, we present some applications of (weak)
hyperstructures to arithmetic functions. Finally, we leave
with some questions and highlight useful steps to solve
some of them.
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1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 3 / 54
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Introduction Motivation
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 4 / 54
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Introduction Motivation
In a chemical reaction, the reaction of two or more elements may result
in more than one element. e.g. The spontaneous redox reaction
At2+At+−→ At++At−.
Mendelian inheritance.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 5 / 54
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Introduction History
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 7 / 54
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Introduction History
In 1934, Frederic Marty, only 23 years old, defined the concept of a
hypergroup at the eighth Congress of Scandinavian Mathematicians [1].
The law characterizing such a structure is called multi-valued operation
or hyperoperation or hypercomposition and the theory of the algebraic
structures endowed with at least one multi-valued operation is known
as the Hyperstructure Theory or Hypercompositional Algebra.
Marty’s motivation to introduce hypergroups is that the
quotient of a group modulo any subgroup (not necessarily
normal) is a hypergroup.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 8 / 54
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Introduction History
It is known that, for a group (G,·)and a subgroup Hof G, the equality
xH ·yH = (xy)Hdefines an operation on the set G/H={xH :x∈G}
of left equivalence classes determined by Hif and only if
x,y∈G,x0∈xH,y0∈yH ⇒x0y0∈(xy)H.
And this happens exactly when His a normal subgroup of G.
In general, xH ?yH ={zH :z=x0y0;x0∈xH,y0∈yH}defines a
function
?:G/H×G/H→ P∗(G/H).
P∗(G/H)denotes the set of non-empty subsets of G/H. Such functions
are called (binary) multi-operations and the corresponding (hyper)structures
are called hypergroups.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 9 / 54
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Introduction History
Illustration
Let (S3={(1),(12),(13),(23),(123),(132)},·)be the symmetric
group on three letters under the usual composition of permutations. i.e.
f.g(n) = f(g(n)) for all f,g∈S3and n∈ {1,2,3}.
A3={(1),(123),(132)}is a normal subgroup of S3. Then
(S3/A3, ?)is a group (quotient group) isomorphic to (Z2,+), the
group of integers modulo 2under standard addition modulo 2.
M={(1),(12)}is a subgroup of S3that is not normal because
M(13) = {(13),(132)} 6= (13)M={(13),(123)}. Then
(S3/M={M,(13)M,(23)M}, ?)is not a group.
This is easily seen as M?(13)M={(13)M,(23)M}/∈S3/M.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 10 / 54
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Introduction History
Illustration (Cont’d)
The hypergroup (S3/M={M,(13)M,(23)M}, ?)is given by the
following table.
?M(13)M(23)M
M M {(13)M,(23)M} {(13)M,(23)M}
(13)M(13)M{M,(23)M} {M,(23)M}
(23)M(23)M{M,(13)M} {M,(13)M}
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 11 / 54
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Introduction History
The theory knew an important progress starting with the 70’s, when its
research area has enlarged. In France, M.Krasner, M.Koskas and
Y.Sureau have investigated the theory of subhypergroups and the
relations defined on hyperstructures; in Greece, J.Mittas,
D.Stratigopoulos, M.Konstantinidou, K.Serafimidis, Ch.G.Massouros
have studied the canonical hypergroups, the hyperrings and the
hypermodules, the hyperlattices, the hyperfields with applications in
Automata Theory.
There are applications of hypergroups in various domains such as:
geometry, topology, analysis of the convex systems, finite groups’
character theory, cryptography and code theory, automata theory,
graphs and hypergraphs, theory of fuzzy and rough sets, theory of
binary relations, probability theory, neutrosophy, and many other fields.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 12 / 54
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Hyperstructure theory Hypergroups
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 13 / 54
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Hyperstructure theory Hypergroups
Definition
Let Hbe a non-empty set. Then, a mapping ◦:H×H→ P∗(H)is
called a binary hyperoperation on H, where P∗(H)is the family of all
non-empty subsets of H. The couple (H,◦)is called a hypergroupoid ([2]).
In this definition, if Aand Bare two non-empty subsets of Hand x∈H,
then we define:
A◦B=S
a∈A
b∈B
a◦b,x◦A={x} ◦ Aand A◦x=A◦ {x}.
Definition
Let (H,◦)be a hypergroupoid. An element e∈His called an identity of
(H,◦)if x∈(x◦e)∩(e◦x), for all x∈Hand it is called a scalar
identity of (H,◦)if x◦e=e◦x={x}, for all x∈H([2]).
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 14 / 54
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Hyperstructure theory Hypergroups
A hypergroupoid (H,◦)is called a:
1semihypergroup if for every x,y,z∈H, we have
x◦(y◦z) = (x◦y)◦z;
2quasi-hypergroup if for every x∈H,x◦H=H=H◦x(The latter
condition is called the reproduction axiom);
3hypergroup if it is a semihypergroup and a quasi-hypergroup.
Definition
A subset Sof a hypergroup (H,◦)is said to be a subhypergroup of Hif
(S,◦)is a hypergroup ([2]).
To prove that Sis a subhypergroup of Hit suffices to show that (S,◦)is a
quasi-hypergroup.
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Hyperstructure theory Hypergroups
Examples
Let Gbe a group. For any x,y∈G, we define x◦y=<x,y>(the
subgroup of Ggenerated by xand y. Then (G,◦)is a hypergroup
([3]).
Let Gbe a group and Hbe a normal subgroup of G. We define on G
the hyperoperation “◦" as x◦y=Hxy and we obtain that (G,◦)is a
hypergroup ([3]).
Biset hypergroup. Let H6=∅be any set and define (H, ?)as follows:
a?b={a,b}for all a,b∈H.
Remark
Every group is a hypergroup (By dealing with the binary operation of
a,b∈Gas the singleton set {ab}.).
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 16 / 54
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Hyperstructure theory Hypergroups
Examples (Cont’d)
Total hypergroup. Let H6=∅be any set and define (H,~)as follows:
a~b=Hfor all a,b∈H.
Let S={a,b,c}. Then (S,)defined by the following table is a
hypergroup ([4]).
a b c
a S {a,b} {a,b}
b{a,b}S{a,c}
c{a,b} {a,c}S
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 17 / 54
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Hyperstructure theory Hypergroups
Some Remarks
1The identity in a hypergroupoid (if it exists) is not necessary unique.
By considering the Biset hypergroup on the set {a,b}, we get that
both: aand bare identities.
2A scalar identity (if it exists) is unique.
3The identity for (S,)is band it is unique although it is not a scalar
identity.
4Unlike groups, a non-trivial hypergroup may be having only one
subhypergroup (which is itself). e.g. (S,)and total hypergroups.
5Like finite groups, finite hypergroups can be presented by means of
Cayley’s table.
6A finite Biset hypergroup of order nhas nsubhypergroups up to
isomorphism.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 18 / 54
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Hyperstructure theory Hyperring
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 19 / 54
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Hyperstructure theory Hyperring
Definition
A multivalued system (R,+,·)is a hyperring if (1) (R,+) is a hypergroup;
(2) (R,·)is a semihypergroup; (3) “·" is distributive with respect to “+"
([2]).
Example 1: Let R1={0,1}. Then (R1,+1,·1), presented by the
following tables, is a hyperring.
+10 1
0 0 1
1 1 R1
and
·10 1
0 0 0
1 0 1
Remark
Every ring is a hyperring.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 20 / 54
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Hyperstructure theory Hyperring
Example 2: [3] Let R2={0,1,2}. Then (R2,+2,·2), presented by the
following tables, is a hyperring.
+20 1 2
0 0 1 2
1 1 1 R2
2 2 R22
and
·2012
0 0 0 0
1 0 1 2
2 0 1 2
Example 3: [3] Let (R,+,·)be a commutative ring. By setting
R={x={−x,x}:x∈R}, we get that (R,,)is a hyperring. Here,
xy={x−y,x+y}
and
xy={x·y}.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 21 / 54
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Hyperstructure theory Hyperring
As an illustration for Example 3, if we take our commutative ring to be
(Z4,+,·)then 0={0},1={1,3},2={2}. Thus, Z4={0,1,2}.
The hyperring (Z4,,)can be presented by the following tables.
0 1 2
0 0 1 2
1 1 {0,2}1
2 2 1 0
and
012
0 0 0 0
1 0 1 2
2 0 2 0
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 22 / 54
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Hvstructures
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 23 / 54
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Hvstructures
Weak hyperstructures or Hv-structures were introduced by Vougiouklis in
the fourth AHA congress in 1990 [5] as a generalization of the well known
algebraic hyperstructures (hypergroup, hyperring, and so on). Hv-structures
are hyperstructures where the equality in some axioms is replaced by the
non-empty intersection.
It is known that the quotient of a group by a normal subgroup of it is a
group and that the quotient of a group by any subgroup of it is a
hypergroup. Vougiouklis stated that the quotient of a group with
respect to any partition of it is an Hv-group.
This class of hyperstructures is very large so one can use it in order to define
several objects that they are not possible to be defined in the classical
hyperstructure theory.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 24 / 54
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Hvstructures
Illustration
Let Z4={0,1,2,3}be the group of integers modulo 4under standard
addition modulo 4with the partition {{0,1},{2},{3}} corresponding to
the equivalence relation R={(0,0),(0,1),(1,0),(1,1),(2,2),(3,3)}.
Then the quotient of Z4with respect to the given partition is
Z4/R={0,2,3}.
(Z4/R, ?)is given by the following table.
?023
0{0,2} {2,3} {0,3}
2{2,3}0 0
3{0,3}0 2
It is clear that (Z4/R, ?)is a quasi-hypergroup.
Since 0?(2?2) = {0,2} 6={0}= (0?2)?2, it follows that (Z4/R, ?)
is not a hypergroup.
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Hvstructures
Definition
A hypergroupoid (H,◦)is called an Hv-semigroup if
(x◦(y◦z)) ∩((x◦y)◦z)6=∅for all x,y,z∈H.
Definition
A hypergroupoid (H,◦)is called an Hv-group if it is an Hv-semigroup and
a quasihypergroup ([3]).
Example 4: (Vougiouklis, 1995) [3] Let (G,·)be a group and Ran
equivalence relation on G. Then (G/R,)is an Hv-group which is not
always a hypergroup. Here, for all equivalence classes x,y∈G/R,
xy={z:z∈x·y}
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 26 / 54
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Hvstructures
Example 5 (Vougiouklis, 1995) [3] (Zmn, ?)is an Hv-group. “?" is
defined as follows:
x?y=({0,m}if (x,y) = (0,m);
x+yotherwise.
As an illustration for Example 5, if we take m=n=2, we get that (Z4, ?)
represented by the following table is an Hv-group.
?0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 {0,2}1
3 3 0 1 2
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 27 / 54
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Hvstructures
Remark
The Hv-group (Zmn, ?)defined in Example 5 is called a very thin Hv-group.
This is because |x?y|>1just in one case.
Example 6: [6] Let H={a,b,c,d}and define (H,⊕)by the following
table:
⊕a b c d
a a {a,c} {a,c}c
b{a,c} {a,c} {a,c}c
c{a,c} {a,c}c{c,d}
d c c {c,d}d
Then (H,⊕)is a commutative Hv-semigroup.
Remark
(H,⊕)is not a semihypergroup nor a quasihypergroup. This is clear as
{c}=b⊕(d⊕d)6= (b⊕d)⊕d={c,d}.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 28 / 54
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Hvstructures
Definition
([3]) A multivalued system (R,+,·)is called an Hv-ring if (1) (R,+) is an
Hv-group; (2) (R,·)is an Hv-semigroup; (3) “·" is weak distributive with
respect to “+" .
(R,+,·)is said to be commutative if x+y=y+xand x·y=y·xfor
all x,y∈R.
Example 7: Let (H, ?)be any Hv-group and define “◦" on Has follows:
x◦y={x,y}for all x,y∈H.
Then (H, ?, ◦)is an Hv-ring.
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 29 / 54
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Hvstructures
As an illustration for Example 7, we get that (H, ?, ◦)represented by the
following tables is a commutative Hv-ring.
?0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 {0,2}1
3 3 0 1 2
◦0 1 2 3
0 0 {0,1} {0,2} {0,3}
1{0,1}1{1,2} {1,3}
2{0,2} {1,2}2{2,3}
3{0,3} {1,3} {2,3}3
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Fundamental relations
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 31 / 54
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Fundamental relations Strongly regular relations
Let (H,◦)be a semihypergroup and Rbe an equivalence relation on H. If
Aand Bare non-empty subsets of H, then ARB means that for every
a∈Aand b∈B, we have aRb ([7]).
The equivalence relation Ris called
1strongly regular on the right (on the left) if for all x∈H, from aRb, it
follows that (a◦x)R(b◦x)((x◦a)R(x◦b)respectively);
2strongly regular if it is strongly regular on the right and on the left.
Theorem
Let (H,◦)be a hypergroup and Ran equivalence relation on H. Then Ris
strongly regular if and only if (H/R,⊗), the set of all equivalence classes,
is a group ([7]).
Madeleine Al-Tahan (LIU) Algebraic Hyperstructures: Applications to Arithmetic FunctionsBIT, June 23rd , 2020 32 / 54
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Fundamental relations Fundamental relations of semihypergroups
The main tools connecting the class of (weak) hyperstructures with the
classical algebraic structures are the fundamental relations. The first who
defined these relations was Koskas in 1970 [8].
Definition
For all n>1, we define the relation βnon a semihypergroup (H,·), as
follows:
aβnb⇐⇒ ∃(x1,...,xn)∈Hn:{a,b} ⊆
n
Y
i=1
xi,
β=Sn≥1βnand β?is the transitive closure of β.
β?is called the fundamental equivalence relation on Hand (H/β?,⊕)is
called the fundamental group.
The operation “⊕" is defined as follows: For all x,y∈H,
β?(x)⊕β?(y) = β?(z)where z∈β?(x)·β?(y).
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Fundamental relations Fundamental relations of semihypergroups
Remark
The relation β∗is the smallest strongly regular relation on H.
Remark
The relation β∗is the smallest equivalence relation on H, such that the
quotient H/β∗is a group. And if His a hypergroup then β∗=β.
Example 8: Let (S,)be the hypergroup defined by the following table.
(S,)defined by the following table is a hypergroup ([4]).
a b c
a S {a,b} {a,b}
b{a,b}S{a,c}
c{a,b} {a,c}S
We have: xβ2yfor all x,y∈Sas x,y∈S=xx. This implies that
xβyfor all x,y∈S. Therefore, S/β?is the trivial group.
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Fundamental relations Fundamental relations of Hv-rings
In 1990, Vougiouklis [5] defined the notion of fundamental relations on
Hv-rings.
Definition
For all n>1, we define the relation γon an Hv-ring (R,+,·)as follows:
aγb⇐⇒ {a,b} ⊆ uwhere uis finite sum of finite products of elements
in R[2].
The relation γis reflexive and symmetric. Denote by γ?the transitive
closure of γ. The γ?is called the fundamental equivalence relation on R
and R/γ?is the fundamental ring.
The operations “⊕" and “⊗" are defined as follows: For all x,y∈R,
γ?(x)⊕γ?(y) = γ?(z)where z∈γ?(x) + γ?(y)
and
γ?(x)⊗γ?(y) = γ?(z)where z∈γ?(x)·γ?(y).
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Hyperstructure theory and arithmetic functions
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
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Hyperstructure theory and arithmetic functions Hyperring of arithmetics
An arithmetic function is a function whose domain is the set of natural
numbers and whose codomain is the set of complex numbers.
Definition (M. Asghari-Larimi and B. Davvaz, 2010)
Let Gbe the set of arithmetic functions. Define a hyperoperation on Gas
follows
◦:G×G→P∗(G)
(α, β)7−→ α◦β
such that (α◦β)(n) = Sd|nα(d)β(n
d)([8]).
Theorem (M. Asghari-Larimi and B. Davvaz, 2010)
Let Gbe the set of arithmetic functions. Then (G,◦)is a commutative
hypergroup ([8]).
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Hyperstructure theory and arithmetic functions Hyperring of arithmetics
Definition (M. Al-Tahan and B. Davvaz, 2017)
Let Gbe the set of arithmetic functions. Define a hyperoperation on Gas
follows
?:G×G→P∗(G)
(α, β)7−→ α ? β
such that
(α ? β)(n) = α(d) + β(n
d) : d|n=[
d|n
α(d) + β(n
d).
Theorem (M. Al-Tahan and B. Davvaz, 2017)
Let Gbe the set of arithmetic functions. Then (G, ?)is a commutative
hypergroup ([9]).
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Hyperstructure theory and arithmetic functions Hyperring of arithmetics
Illustration
Let α(n) = n,β(n) = 1,γ1(n) = 2,γ2(n) = n+1and
γ3(n) = (2if n=1or n=2
n+2otherwise. .
Having (α ? β)(n) = {1+d:d|n}implies that γ1, γ2, γ3∈α ? β.
Having (α◦β)(n) = {d:d|n}implies that α, β ∈α◦β.
(α ? β)(1) = {2},(α ? β)(2) = {2,3},(α ? β)(3) = {2,4},
(α ? β)(4) = {2,3,5},(α ? β)(5) = {2,6},. . .
For γto be in α ? β,γ(n)∈(α ? β)(n). So, for each value of kwe
choose for γ(k)a value from (α?β)(k). Thus, γis an arithmetic function
and hence, α ? β ⊆G.
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Hyperstructure theory and arithmetic functions Hyperring of arithmetics
The identity 0?for (G, ?)is defined as follows:
0?(n) = 0for all n∈N.
It is clear α∈(α ? 0?)∩(0?? α)as
(0?? α)(n) = (α ? 0?)(n) = {α(d) : d|n}.
Remark
The identity ıfor (G,◦)is unique and its is defined as follows:
ı(n) = (1if n=1
0otherwise.
This is clear as (α◦ı)(n) = {α(n),0}. So, α∈(α◦ı)∩(ı◦α)for all
α∈G.
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Hyperstructure theory and arithmetic functions Hyperring of arithmetics
Lemma (M. Al-Tahan and B. Davvaz, 2017)
Let AF (G)be the set of all additive arithmetic functions. Then
(AF (G), ?)is a subhypergroup of (G, ?)([9]).
Theorem (M. Al-Tahan and B. Davvaz, 2017)
Let Gbe the set of arithmetic functions. Then (G, ?, ◦)is a commutative
Hv-ring ([9]).
Remark
Let Gbe the set of arithmetic functions. Then (G, ?, ◦)is not a hyperring.
Theorem (M.Al-Tahan and B. Davvaz, 2017)
Let Mbe the set of all constant arithmetic functions. Then the largest
hyperring contained in (G, ?, ◦)is (M, ?, ◦)([9]).
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Hyperstructure theory and arithmetic functions Fundamental group of the hypergroup of arithmetics
Theorem (M. Al-Tahan and B. Davvaz, 2018)
Let (G, ?)be the hypergroup of arithmetic functions, α, γ ∈G. Then
αβγ ⇐⇒ α(1) = γ(1).
Theorem (M. Al-Tahan and B. Davvaz, 2018)
Let (G, ?)be the hypergroup of arithmetic functions and β=β?the
fundamental relation on G. Then
(G/β, ⊕)∼
=(C,+).
Proof.
Let ψ: (G/β, ⊗)→(C,+) be defined as ψ(β(α)) = α(1). By showing
that ψis a group isomorphism and using the fact that βand β?coincide in
this case as (G, ?)is a hypergroup, we get the result.
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Hyperstructure theory and arithmetic functions Fundamental group of the hypergroup of arithmetics
What can we deduce?
The fundamental group of the hypergroup of arithmetics is
the group of complex numbers under standard addition (up
to isomorphism).
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Hyperstructure theory and arithmetic functions Fundamental ring of the Hv-ring of arithmetics
Theorem (M. Al-Tahan and B. Davvaz, 2018)
Let (G, ?, ◦)be the Hv-ring of arithmetic functions and γ?the
fundamental relation on G. Then
γ?=β?.
Here, β?=βis the fundamental relation of the hypergroup (G, ?).
Moreover, (G/γ?,⊕,⊗)∼
=(C,+, .)([10]).
Proof.
Having G/γ?a ring implies that (G/γ ?,⊕)is a group. Since (G, ?)is a
hypergroup and γ?is an equivalence relation, it follows that γ?is a strongly
regular relation on (G, ?). Define χ: (G/β?,⊕,⊗)−→ (C,+,·)by
χ(α) = α(1). Since χis a ring isomorphism, it follows that (G/β?,⊕,⊗)
is a ring. Thus, γ?=β?
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Hyperstructure theory and arithmetic functions Fundamental ring of the Hv-ring of arithmetics
What can we deduce?
The fundamental ring of the Hv-ring of arithmetics is the
ring of complex numbers under standard addition and
multiplication (up to isomorphism).
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Conclusion
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
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Conclusion
In this talk, we were concerned about only some hyperstructures
(hypergroups of Marty and hyperrings) as well as some Hv-structures
(Hv-groups(semigroups) and Hv-rings). We presented some applications of
these (weak)hyperstructures to arithmetic functions by discussing a
hypergroup of arithmetic functions as well as an Hv-ring of arithmetic
functions. Also, we found the fundamental group of arithmetic functions (in
the sense of Koskas) and the fundamental ring of arithmetics (in the sense
of Vougiouklis).
There are many other defined types of (weak)hyperstructures that weren’t
dicussed in this talk. e.g. canonical hypergroups, (weak)polygroups,
semihyperrings, Krasner hyperrings, multiplicative hyperrings, hypermodules
(Hv-modules), hypervector spaces (Hv-vector spaces), etc.
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Conclusion
Different fundamental relations were defined on (weak)hyperstructures. For
example, Freni in 2002 [11] defined an equivalence relation on
semihypergroups and proved that the quotient is an abelian group. Also, he
proved that it is the smallest strongly regular relation such that the quotient
is an abelian group.
In our work, our relation and the relation defined by Freni coincide. This is
because our hypergroup is commutative. Here, we can raise some questions.
1Can we find a necessary and a sufficient condition for the two
relations to coincide?
2Is the fundamental abelian group, up to isomorphism, a subgroup
of the fundamental group?
3Can we find a strongly equivalence relation Ron the hypergroup
(G, ?)such that (G/R,⊕)is a cyclic group?
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Conclusion
To answer the last question, we need the following steps.
1Define an equivalence relation on semihypergroups in a way that the
quotient is a cyclic group.
2Use the fact that a cyclic group is, up to isomorphism, either the trivial
group or (Z,+) or (Zn,+).
3Use our characterization in [10] for all strongly regular relations on
(G, ?)and for (G/R,⊕). These characterizations may help in solving
this problem.
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References
1Introduction
Motivation
History
2Hyperstructure theory
Hypergroups
Hyperring
3Hvstructures
4Fundamental relations
Strongly regular relations
Fundamental relations of semihypergroups
Fundamental relations of Hv-rings
5Hyperstructure theory and arithmetic functions
Hyperring of arithmetics
Fundamental group of the hypergroup of arithmetics
Fundamental ring of the Hv-ring of arithmetics
6Conclusion
7References
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References
F. Marty, Sur une generalization de la notion de group, In 8th Congress
Math., Scandenaves, Stockholm, Sweden, 14-18 August 1934, 45-49.
B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications,
International Academic Press, USA, 2007.
B. Davvaz, I. Cristea, Fuzzy Algebraic Hyperstructures, Studies in
Fuzziness and Soft Computing 321, Springer International Publishing
2015.
M. Al-Tahan and B. Davvaz, Commutative single power cyclic
hypergroups of order three and period two, Discrete Mathematics,
Algorithms and Applications 9(5)(2017).
T. Vougiouklis, The fundamental relation in hyperrings, In: The general
hyperfield: Algebraic hyperstructures and applications, Proc. Fourth Int.
Congress on Algebraic Hyperstructures and Appl. (AHA 1990), Xanthi,
1990, 203-211.
M. Al- Tahan, B. Davvaz, Chemical hyperstructures for Astatine,
Tellurium and for Bismuth, Bull. Comput. Math., 7(1)(2019) 9-25.
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References
P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore,
Tricesimo, 1993.
M. Koskas, Groupoides, demi-hypergroupes et hypergroupes, J. Math.
Pures Appl. 49(1970) 155-192.
M. Asghari-Larimi and B. Davvaz, Hyperstructures associated to
arithmetic functions, Ars Combin. 97(2010) 51-63
M. Al-Tahan and B. Davvaz, On the existence of hyperrings associated
to arithmetic functions, J. Number Theory 174(2017) 136-149.
M. Al-Tahan and B. Davvaz, Strongly regular relations of arithmetic
functions, J. Number Theory 187(2018) 391-402.
D. Freni, A new characterization of the derived hypergroup via strongly
regular equivalences, Communication in Algebra 30(8)(2002)3977-3989.
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