Symmetry in Hyperstructure: Neutrosophic Extended Triplet Semihypergroups and Regular Hypergroups

Xiaohong Zhang,Florentin Smarandache,Yingcang Ma

doi:10.3390/sym11101217

Abstract

The symmetry of hyperoperation is expressed by hypergroup, more extensive hyperalgebraic structures than hypergroups are studied in this paper. The new concepts of neutrosophic extended triplet semihypergroup (NET- semihypergroup) and neutrosophic extended triplet hypergroup (NET-hypergroup) are firstly introduced, some basic properties are obtained, and the relationships among NET- semihypergroups, regular semihypergroups, NET-hypergroups and regular hypergroups are systematically are investigated. Moreover, pure NET-semihypergroup and pure NET-hypergroup are investigated, and a strucuture theorem of commutative pure NET-semihypergroup is established. Finally, a new notion of weak commutative NET-semihypergroup is proposed, some important examples are obtained by software MATLAB, and the following important result is proved: every pure and weak commutative NET-semihypergroup is a disjoint union of some regular hypergroups which are its subhypergroups.

Highlights

Introduction and PreliminariesAs a generalization of traditional algebraic structures, hyper algebraic structures have been extensively studied and applied [1,2,3,4,5,6,7]
We propose a new notion of weak commutative NET-semihypergroup, and prove the structure theorem of weak commutative pure NET-semihypergroup (WCP-NET-semihypergroup), which can be regarded as a generalization of Cliffod Theorem in semigroup theory
We propose some new notions of neutrosophic extended triplet semihypergroup (NET-semihypergroup), neutrosophic extended triplet hypergroup (NET-hypergroup), pure NETsemihypergroup and weak commutative NET-semihypergroup, investigate some basic properties and the relationships among them, study their close connections with regular hypergroups and regular semihypergroups

Summary

Introduction and Preliminaries

As a generalization of traditional algebraic structures, hyper algebraic structures (or hypercompositional structures) have been extensively studied and applied [1,2,3,4,5,6,7]. Hypergroups and semihypergroups are basic hyper structures which are extensions of groups and semigroups [8]. N is called a neutrosophic extended triplet set (NETS). If (N,*) is a semigroup, (N, *) is called to be a neutrosophic extended triplet group (NETG). Let (H,*) be a semihypergroup (i.e., * be a binary hyperoperation on nonempty set H such that (x*y)*z = x*(y*z), for all x, y, z∈H). (H,*) is called a neutrosophic extended triplet semihypergroup (shortened form, NET-semihypergroup), if for every x∈H, there exist neut(x) and anti(x) such that x∈(neut(x)*x)∩(x*neut(x)), and Definition 7. Let (H,*) be a semihypergroup (i.e., * be a binary hyperoperation on nonempty set H such that neut(x)∈(anti(x)*x)∩(x*anti(x)). (H,*) is called a neutrosophic extended triplet semihypergroup (shortened form, NET-semihypergroup), if for every x∈H, there exist neut(x) and anti(x) such that.

Verify

Pure NET-semihypergroups and Regular hypergroups

Weak Commutative NET-Semihypergroups and Their Structures

Conclusions