Ternary Semigroups Research Articles

On Prime Ideal Space of a Partially Ordered Ternary Semigroup

In this paper, we introduced the hull-kernel topology $\tau$ on the set $\mathcal P$ of prime ideals in a partially ordered ternary semigroup $T$ and investigated various topological properties of the structure space $(\mathcal P, \tau)$. We also obtained some useful results about compactness and connectedness of the set of all prime full ideals of $T$.

A study of bi-bases of ternary semigroups

A study of bi-bases of ternary semigroups

Almost Bi-ideals and Fuzzy Almost Bi-ideals of Ternary Semigroups

Almost Bi-ideals and Fuzzy Almost Bi-ideals of Ternary Semigroups

General types of sup -hesitant fuzzy ideals of ternary semigroups

General types of sup -hesitant fuzzy ideals of ternary semigroups

Hybrid interior ideals and hybrid bi-ideals in ternary semigroups

This article explores the concept of hybrid interior ideals and the hybrid characteristic interior ideals of a ternary semigroup. We establish some equivalent requirements for a hybrid structure to be a hybrid interior ideal of a ternary semigroup. Also, we show that for a regular semigroup and an intra-regular semigroup, hybrid interior ideals and hybrid ideals coincide. Furthermore, the notion of a hybrid bi-ideal of a ternary semigroup is introduced, and some of its key characteristics are explored.

TERNARY ∗-BANDS ARE GLOBALLY DETERMINED

A non-empty set \(S\) together with the ternary operation denoted by juxtaposition is said to be ternary semigroup if it satisfies the associativity property \(ab(cde)=a(bcd)e=(abc)de\) for all \(a,b,c,d,e\in S\). The global set of a ternary semigroup \(S\) is the set of all non empty subsets of \(S\) and it is denoted by \(P(S)\). If \(S\) is a ternary semigroup then \(P(S)\) is also a ternary semigroup with a naturally defined ternary multiplication. A natural question arises: "Do all properties of \(S\) remain the same in \(P(S)\)?" The global determinism problem is a part of this question. A class \(K\) of ternary semigroups is said to be globally determined if for any two ternary semigroups \(S_1\) and \(S_2\) of \(K\), \(P(S_1)\cong P(S_2)\) implies that \(S_1\cong S_2\). So it is interesting to find the class of ternary semigroups which are globally determined. Here we will study the global determinism of ternary \(\ast\)-band.

$\inf$-Hesitant and $(\sup, \inf)$-Hesitant fuzzy ideals of ternary semigroups

The notions of an $\inf$-hesitant fuzzy right (left, lateral) ideal, an $\inf$-hesitant fuzzy ideal, a $(\sup, \inf)$-hesitant fuzzy right (left, lateral) ideal, and a $(\sup, \inf)$-hesitant fuzzy ideal, which are generalizations of an interval-valued fuzzy ideal of a ternary semigroup, are introduced and their properties are investigated. Conditions for a hesitant fuzzy set to be an $\inf$-hesitant fuzzy right (left, lateral) ideal, an $\inf$-hesitant fuzzy ideal, a $(\sup, \inf)$-hesitant fuzzy right (left, lateral) ideal, and a $(\sup, \inf)$-hesitant fuzzy ideal of a ternary semigroup are provided in terms of sets, fuzzy sets, Pythagorean fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets. Furthermore, characterizations of an ideal of a ternary semigroup are studied via a generalization of the characteristic hesitant and the characteristic interval-valued fuzzy set.

On right bases of partially ordered ternary semigroups

We investigate the results of a partially ordered ternary semigroup containing right bases and characterize when a non-empty subset of a partially ordered ternary semigroup is a right base. Moreover, we give a characterization of a right base of a partially ordered ternary semigroup to be a ternary subsemigroup and we show that the right bases of a partially ordered ternary semigroup have same cardinality. Finally, we show that the complement of the union of all right bases of a partially ordered ternary semigroup is a maximal proper left ideal.

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.

Prime Quasi-Ideals in Ternary Seminear Rings

Objectives/Background: The ternary seminear ring is the generalization of seminear ring and it need not be a ternary semiring. Characterization of quotient ternary seminear rings and some structures of ternary seminear ring have been analysed and also studied ideals in ternary seminear rings. Further quasi ideals in ternary seminear rings defined and discussed about their properties. Methods: Properties of seminear ring and ternary semiring have been employed to carry out this research work to obtain all the characterizations of ternary seminear rings corresponding to that ternary semiring. Findings: We call an algebraic structure (T;+; :) is a ternary seminear ring if (T,+) is a Semigroup, T is a ternary semigroup under ternary multiplication and xy(z+u) = xyx+xyu for all x;y; z;u 2 T. T is said to have an absorbing zero if there exists an element 0 2 T such that x+0 = 0+x = x for all x 2 T and xy0 = x0y = 0xy = 0 for all x;y 2 T. Throughout this paper T will always stand for ternary seminear ring with an absorbing zero. In this ternary structure we try to study prime quasi ideals concept and obtain their properties. Novelty: In this study, we define the notion of Prime quasi ideals in ternary seminear rings. We also find some of their interesting results. AMS Subject Classification code: 16Y30,16Y99,17A40 Keywords: Ternary seminear ring; Idempotent ternary seminear ring; Ideals in ternary seminear ring; Quasi Ideals in ternary seminear ring; Prime Ideals in ternary seminear rings.

Lattice structures in ternary semigroup of mappings

Lattice structures in ternary semigroup of mappings

Applications of Spherical Fuzzy Sets in Ternary Semigroups

In this paper, we introduce the notions of spherical fuzzy ternary subsemigroups and spherical fuzzy ideals in ternary semigroups by using the concepts of ternary subsemigroups and ideals in ternary semigroups. We investigate their properties. Moreover, we study roughness of spherical fuzzy ideals in ternary semigroups.

Rough Fuzzy Ideals Induced by Set-Valued Homomorphism in Ternary Semigroups

The main objective of this paper is to characterize rough approximations of fuzzy ideals in ternary semigroups. Rough fuzzy ideals are used to deal with vague and incomplete information in decision-making problems. In this research, approximations for fuzzy prime ideals in ternary semigroups are studied. It is proved that generalized lower approximations and generalized upper approximations of ∈ , ∈ ∨ q -fuzzy prime (resp., semiprime) ideals of ternary semigroups are ∈ , ∈ ∨ q -fuzzy prime (resp., semiprime) ideals. For this, the concept of SSVH (strong set-valued homomorphism) and SVH (set-valued homomorphism) is used. Also, it is shown by examples that the lower approximations of fuzzy subsemigroups and fuzzy ideals are not fuzzy subsemigroups and fuzzy ideals, respectively, for SVH.

On irreducible pseudo symmetric ideals of a partially ordered ternary semigroup

In this paper, the concepts of irreducible and strongly irreducible pseudo symmetric ideals in a partially ordered ternary semigroup are introduced. We also studied some interesting properties of irreducible and strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup and prove that the space of strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup is topologized.

Characterization of almost ternary subsemigroups and their fuzzifications

In this paper, we define almost ternary subsemigroups of ternary semigroups. Almost ternary subsemigroups is a one of generalizations of ternary subsemigroups. We show that the union of two almost ternary subsemigroups is also an almost ternary subsemigroup. However, the intersection of two almost ternary subsemigroups need not be an almost ternary subsemigroup. Moreover, we define their fuzzifications and provide the relationships between almost ternary subsemigroups and their fuzzifications.

Ideals in topological ternary semigroups

Ideals in topological ternary semigroups

A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment

<abstract> <p>Theory of $m$-polar fuzzy set deals with multi-polar information. It is used when data comes from $m$ factors $\left({m \ge 2} \right)$. The primary objective of this work is to explore a generalized form of $m$-polar fuzzy subsemigroups, which is $m$-polar fuzzy ternary subsemigroups. There are many algebraic structures which are not closed under binary multiplication that is a reason to study ternary operation of multiplication such as the set of negative integer is closed under the operation of ternary multiplication but not closed for the binary multiplication. This paper, presents several significant results related to the notions of $m$-polar fuzzy ternary subsemigroups, $m$-polar fuzzy ideals, $m$-polar fuzzy generalized bi-ideals, $m$-polar fuzzy bi-ideals, $m$-polar fuzzy quasi-ideals and $m$-polar fuzzy interior ideals in ternary semigroups. Also, it is proved that every $m$- polar fuzzy bi-ideal of ternary semigroup is an $m$-polar fuzzy generalized bi-ideal of ternary semigroup but converse is not true in general. Moreover, this paper characterizes regular and intra-regular ternary semigroups by the properties of $m$-polar fuzzy ideals, $m$-polar fuzzy bi-ideals.</p> </abstract>

Radicals of generalized prime ideals in ternary semigroups

Radicals of generalized prime ideals in ternary semigroups

Ordered power ternary semigroups

We consider a power ternary semigroup [Formula: see text] associated with a ternary semigroup [Formula: see text] and study some properties of [Formula: see text] by using the corresponding properties of [Formula: see text]. After that we study the notion of ordered power ternary semigroup and our main aim is to establish some interconnection between the properties of a ternary semigroup [Formula: see text] and the associated ordered ternary semigroup [Formula: see text].

V-Regular Ternary Menger Algebras and Left Translations of Ternary Menger Algebras

Let n be a fixed natural number. Ternary Menger algebras of rank n, which was established by the authors, can be regarded as a suitable generalization of ternary semigroups. In this article, we introduce the notion of v-regular ternary Menger algebras of rank n, which can be considered as a generalization of regular ternary semigroups. Moreover, we investigate some of its interesting properties. Based on the concept of n-place functions (n-ary operations), these lead us to construct ternary Menger algebras of rank n of all full n-place functions. Finally, we study a special class of full n-place functions, the so-called left translations. In particular, we investigate a relationship between the concept of full n-place functions and left translations.