Γ-Semihypergroups and Regular Relations

Abstract

In this paper we discuss the structure of Γ-semihypergroups. We prove some basic results and present several examples of Γ-semihypergroups. Also, we obtain some properties of regular and strongly regular relations on a Γ-semihypergroup and construct a Γ-semigroup from a Γ-semihypergroup by using the notion of fundamental relation. Introductione algebraic hyperstructure notion was introduced in 1934 by the French mathematician Marty [1], at the 8th Congress of Scandinavian Mathematicians. He published some notes on hypergroups, using them in different contexts: algebraic functions, rational fractions, and noncommutative groups. Algebraic hyperstructures are a suitable generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set.Around the 1940s, the general aspects of the theory, the connections with groups, and various applications in geometry were studied. e theory knew an important progress starting with the 1970s, when its research area enlarged. A recent book on hyperstructures [2] points out to their applications in cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Basic De�nitionsIn this section, we recall certain de�nitions and results needed for our purpose.Let and be two nonempty sets, the set of all mapping from to , and Γ a set of some mappings from to . e usual composition of two elements of cannot be de�ned. But if we take from and from Γ, then the usual mapping composition can be de�ned. Also, we see that and ( ( for and Γ. Sen and Saha [12] de�ned the notion of a Γ-semigroup as a generalization of a semigroup as follows.De�nition 1� Let and Γ be two nonempty sets. en is called a Γ-semigroup if there exists a mapping Γ written as ( satisfying the following identity:Let be a nonempty subset of . en is called a sub-Γ-semigroup of if for all and Γ. where ∈ and ∈ Γ. An algebraic system that satisfying the associativity property of the above type is a Γ-semigroup.Example 3. Let = and Γ = { / i a positive integer}. en is a gamma-semigroup under the usual multiplication. Next, let = . We have as a nonempty subset of and ∈ for all ∈ and ∈ Γ. en is a sub-Γ-semigroup of .More examples of Γ-semigroup can be found in [12,16]. Let be a nonempty set and let ℘ * ( be the set of all nonempty subsets of . A hyperoperation on is a map ∘ ∶ ℘ * ( and the couple ( ∘ is called a hypergroupoid. If and are non-empty subsets of , then we denote ∘ = ⋃ ∈ ∈ ∘ , ∘ = { } ∘ andof one has ( ∘ ∘ = ∘ ( ∘ , which means thatA semihypergroup ( ∘ is called a hypergroup if for all ∈ , one has ∘ = ∘ = (4) -SemihypergroupsIn this section, we introduce the notion of a Γ-semihypergroup which is a generalization of the notion of Γ-semigroup and semihypergroup.�e�nition �. Let and Γ be two non-empty sets. is called a Γ-semihypergroup if every ∈ Γ be a hyperoperation on , that is, for every ∈ , and for every ∈ Γ and ∈ one has ( = ...

Structures of bipolar fuzzy Γ-hyperideals in Γ-semihypergroups

“…Several authors [28][29][30][31][32][33] studied different properties of -semihypergroup which is a generalization of semigroup, semihypergroup and -semigroup. Fuzzy set theory has been well developed in the context of hyperalgebraic structure theory [34].…”

Fuzzy Γ-hypergroups

“…A fuzzy -hyperoperation assigns to every pair of elements of H and every element of a nonzero fuzzy set on H, while a -hyperoperation assigns to every pair of elements of H and every element of a non-empty subset of H. -hyperstructures represent a generalization of classical algebraic structures and also a generalization of algebraic hyperperstructures, [17,27]. This paper presents an algebraic study of fuzzyhypergroups and fuzzy -hypergroup homomorphisms in connections with the associated -hypergroups and -hypergroup homomorphisms, respectively.…”

“…Let us mention two examples from [27]. Many other examples of -semihypergroups and -hypergroups can be found in [17,27].…”

“…Finally, let us recall what regular and strongly regular relations on -hypergroups are (see [27]). If H is a -semihypergroup and ρ ⊆ H × H is an equivalence relation on H, then for all pairs (A, B) of non-empty subsets of H, we set AρB if and only if for every a ∈ A there exists b ∈ B such that aρb and for every b ∈ B there exists a ∈ A such that aρb.…”

Some properties of hyperideals in ternary semihypergroups

“…al. [2,29,31,48] introduced the notion of Γ-semihypergroup as a generalization of a semigroup, a generalization of a semihypergroup and a generalization of a Γ-semigroup. They presented many interesting examples and obtained a several characterizations of Γ-semihypergroups.…”