Ternary Semihypergroups Research Articles

A novel approach to study ternary semihypergroups in terms of prime soft hyperideals

<abstract> <p>In this paper, we give the generalized form of soft semihypergroups in ternary structure and have studied it with the help of examples. There are some structures that are not appropriately handled by using the binary operation of the semihypergroup, such as all the sets of non-positive numbers are not closed under binary operation but hold for ternary operation. To deal with this type of problem and handling special type of uncertainty, we study the ternary semihypergroup in terms of prime soft hyperideals. We have introduced prime, strongly prime, semiprime, irreducible and strongly irreducible soft bi-hyperideals in ternary semihypergroups and studied certain properties of these soft bi-hyperideals in ternary semihypergroups. The main advantage of this paper is that we proved that each soft bi-hyperideal of ternary semihypergroup $K$ is strongly prime if it is idempotent and the set of soft bi-hyperideals of $K$ is totally ordered by inclusion.</p> </abstract>

Semipseudosymmetric hyperideals in ternary semihypergroups

The main objective of this study is to introduce a new class of hyperideals called pseudosymmetric and semipseudosymmetric hyperideals in ternary semihypergroups. We try to investigate the properties of these hyperideals by giving some suitable examples for the same. In particular we prove that every pseudosymmetric hyperideal of a ternary semihypergroups is a semipseudosymmetric hyperideal. We study the fundamental relations among these hyperideals and provide few results.

Representations, Translations and Reductions for Ternary Semihypergroups

The concept of ternary semihypergroups can be considered as a natural generalization of arbitrary ternary semigroups. In fact, each ternary semigroup can be constructed to a ternary semihypergroup. In this article, we investigate some interesting algebraic properties of ternary semihypergroups induced by semihypergroups. Then, we extend the well-known result on group theory and semigroup theory, the so-called Cayley’s theorem, to study on ternary semihypergroups. This leads us to construct the ternary semihypergroups of all multivalued full binary functions. In particular, we investigate that each element of a ternary semihypergroup induced by a semihypergroup can be represented by a multivalued full binary function. Moreover, we introduce the concept of translations for ternary semihypergroups which can be considered as a generalization of translations on ternary semigrgoups. Then, we construct ternary semihypergroups of all multivalued full functions and ternary semihypergroups via translations. So, some interesting algebraic properties are investigated. At the last section, we discover that there are ternary semihypergroups satisfying some significant conditions which can be reduced to semihypergroups. Furthermore, ternary semihypergroups with another one condition can be reduced to idempotent semihypergroups.

Study of Properties of Several Kinds of Hyperideals in Ordered Ternary Semihypergroups

A non-empty subset S together with an associative function f from S × S×S into the family of all non-empty subsets of S is called a ternary semihypergroup. In this paper, we consider a semihypergroup (S, f) besides a binary relation ≤, where ≤ is a partial order relation on S such that satisfies the monotone condition. This structure is called an ordered ternary semihypergroup. We introduce and investigate the notions of bi-hyperideal and quasi-hyperideal in ordered ternary semihyperroups. In particular, we prove that an ordered ternary semihypergroup is left and right simple if and only if it does not contain proper bi-hyperideals

Int-soft bi-hyperideals in ordered ternary semihypergroups

The aim of this paper is to study po-ternary semihypergroups in terms of the int-soft bi-hyperideals. We introduce the notion of int-soft bi-hyperideals in po-ternary semihypergroups and some properties of them are investigated. Characterizations of bi-hyperideals in terms of int-soft bi-hyperideals are obtained. We prove that every int-soft hyperideal is an int-soft bi-hyperideal but the converse is not true. Examples are provided to illustrate the results.

On Pythagorean normal subbisemiring of bisemiring

In the present paper, we introduce the relative left, right, lateral, two-sided hyperideal, relative quasi-hyperideal, relative bi-hyperideal, relative sub-idempotent ordered bi-hyperideal, relative generalized quasi-hyperideal, relative generalized bi-hyperideal, relative regularity of ordered ternary semihypergroups and relative left (right, lateral) simple ordered ternary semihypergroups. We characterize relative regular ordered ternary semihypergroups through relative quasi-hyperideals and relative bi-hyperideals. We also obtain some results based on relative simple ordered ternary semihypergroups, and other results connecting these relative hyperideal-theoretic notions.

Green’s relations on ternary semihypergroups and crossed hyperproduct of hypergroups

In this paper, we introduce all Green’s relations on the ternary semihypergroup and study some properties of these equivalence relations in view of those obtained in binary and ternary semigroups. We also define crossed hyperproduct of two hypergroups.

On Minimal and Maximal Hyperidealsin n-ary Semihypergroups

The concept of j-hyperideals, for all positive integers 1≤j≤n and n≥2, in n-ary semihypergroups, is a generalization of the concept of left, lateral and right hyperideals in ternary semihypergroups. In this paper, we first introduce the concept of j-(0-)simple n-ary semihypergroups and discuss their related properties through terms of j-hyperideals. Furthermore, we characterize the minimality and maximality of j-hyperideals in n-ary semihypergroups and establish the relationships between the (0-)minimal, maximal j-hyperideals and the j-(0-)simple n-ary semihypergroups. Our main results are to extend and generalize the results on semihypergroups and ternary semihypergroups. Moreover, a related question raised by Petchkaew and Chinram is solved.

A new approach towards int-soft hyperideals in ordered ternary semihypergroups

This paper deals with a class of algebraic hyperstructures called ordered ternary semihypergroups which are studied in terms of int-soft hyperideals. We introduce the notion of int-soft hyperideals in ordered ternary semihypergroups and investigate some properties of them. We also introduce the concepts of convex soft sets and int-soft points and discuss their properties. Moreover, the classes of regular and intra-regular ordered ternary semihpergroups are characterized in terms of int-soft hyperideals.

Completely prime hyperideals of ternary semihypergroups

Completely prime hyperideals of ternary semihypergroups

On soft ordered ternary semihypergroups

Theory of soft sets proposed by Molodtsov as a general framework for reasoning about vague concepts is a powerful mathematical tool for modelling various types of uncertainties. In this paper, we introduce the notions of intersectional soft subsemihypergroup, intersectional soft left (lateral, right) hyperideal of ordered ternary semihypergroups and related properties are investigated. We present characterizations of right weakly regular ordered ternary semihypergroups by means of intersectional hyperideals.

Semihypergroups associated with ternary relations

Davvaz and Leoreanu-Fotea (Commun Algebra 38(10):3621–3636, 2010) studied binary relations on ternary semihypergroups. A ternary relation or triadic relation is a relation in which the number of places in the relation is three. Now, in this paper, instead of binary relations we consider ternary relations and instead of ternary semihypergroups we consider ordinary semihypergroups. Then, we study ternary relations on semihypergroups. In particular, we discuss some properties of compatible ternary relations on them.

Ternary semihypergroups in terms of generalized quasi(bi)-hyperideals

This paper deals with a class of algebraic hyperstructures called ternary semihypergroups. In this paper, we introduce the notion of generalized quasi (bi)-hyperideals in ternary semihypergroups and study their structure. Some related properties of them are investigated. Several characterizations of ternary semihypergroups in terms of minimal generalized quasi(bi)-hyperideals are provided. Also, the n-left simple, m-right simple, (p, q)-lateral simple and (m, (p, q), n)-quasi-simple ternary semihypergroups are defined and investigated.

On the structure of quasi-hyperideals and bi-hyperideals in ternary semihypergroups

This paper deals with a class of algebraic hyperstructures called ternary semihypergroups, which are a generalization of ternary semigroups. Our main purpose of this paper is to introduce and study the notions of quasi-hyperideal and bi-hyperideal in ternary semihypergroups. The structure of ternary semihypergroups through the quasi- and bi-hyperideals is studied and some characterizations are given. Also, we introduce the notion of generalized bi-hyperideals. Several kinds of them are investigated.

Some properties of hyperideals in ternary semihypergroups

Abstract Ternary semihypergroups are algebraic structures with one ternary associative hyperoperation. In this paper we give some properties of left (right) and lateral hyperideals in ternary semihypergroups. We introduce the notion of left simple, lateral simple, left (0-)simple and lateral 0-simple ternary semihypergroups and characterize the minimality and maximality of left (right) and lateral hyperideals in ternary semihypergroups. The relationship between them is investigated in ternary semihypergroups extending and generalizing the analogues results for ternary semigroups.

On intra-regular ternary semihypergroups

In this paper we give some characterizations of the intra-regular ternary semihypergroups in terms of bi-hyperideals and quasi-hyperideals, bi-hyperideals and left hyperideals, bi-hyperideals and right hyperideals of ternary semihypergroups.

Rough Fuzzy Hyperideals in Ternary Semihypergroups

The relations between rough sets and algebraic systems have been already considered by many mathematicians, and rough sets have been studied in various kinds of algebraic systems. This paper concerns a relationship between rough sets and ternary semihypergroups. We introduce the notion of rough hyperideals and rough bi-hyperideals in ternary semihypergroups. We also study fuzzy, rough, and rough fuzzy ternary subsemihypergroups (left hyperideals, right hyperideals, lateral hyperideals, hyperideals, and bi-hyperideals) of ternary semihypergroups.

Binary Relations on Ternary Semihypergroups

Ternary semihypergroups are algebraic structures with one associative hyperoperation. The main propose of this article is to study binary relations on ternary semihypergroups and study some basic properties of compatible relations on them. In particular, we analyze the ternary hypergroup associated with a binary relation.