Roughness and Fuzziness in Ordered Ternary Semigroups

In this paper we introduced the notions of rough left (right, lateral) ideal and rough prime ideals in ternary semigroup, and studied some properties of these ideals. ρ − (A) = {x ∈ S : [x] ρ ⊆ A}, ρ¯(A) = {x ∈ S : [x] ρ ∩ A = ∅} called ρ-lower and ρ-upper approximations of A, respectively.

Information Sciences, 2014

In this paper we initiated the study of roughness in ordered semigroups based on pseudoorder. We introduced the notions of upper (lower) rough ideal, bi-ideal, prime ideal and also the notions of upper (lower) rough fuzzy ideal, rough prime fuzzy ideal in ordered semigroups. We studied some properties of such ideals. Key words and phrases. Upper (lower) rough ideal, upper (lower) rough bi-ideal, upper (lower) rough fuzzy ideal, upper (lower) rough prime fuzzy ideal. ______________________________________________________________________________ 1. Preliminaries An ordered semigroup () ,, S ⋅≤ , is a poset (,) S ≤ , at the same time a semigroup (,) S ⋅ such that ab ≤ implies axbx ≤ and xa ≤ xb for all ,, abxS ∈. A non-empty subset A of an ordered semigroup S is called a subsemigroup of S if 2 AA ⊆. A non-empty subset I of an ordered semigroup S is called a right (resp. left) ideal of S if ISI ⊆ (resp. SII ⊆) and aI ∈ and bS ∈ such that ba ≤ implies bI ∈. I is called an ideal of S if it is both a right and a left ideal of S. An ideal P of an ordered semigroup S is called a prime ideal if xyP ∈ implies xP ∈ or yP ∈ , for all , xyS ∈ [ ] 8. A subsemigroup B of an ordered semigroup S is called a bi-ideal of S if BSBB ⊆ and aB ∈ and bS ∈ such that ba ≤ implies bB ∈ .

In this paper we introduce the notion of fuzzy interior ideals in ternary semigroups and investigated relations between fuzzy ideals and fuzzy interior ideals in terms of regularity. Here a characterization of fuzzy interior ideals is obtained in terms of fuzzy translation operator. The notions of rough and rough fuzzy interior ideals in a ternary semigroup are introduced. Relation between congruences and rough interior ideals are also established here.

2013

Abstract: The concept of, q-fuzzy ternary subsemigroup (left, right, lateral, quasi-, bi-) ideal of a ternary semigroup is introduced and several related properties are investigated. We give some characterizations of regular and weakly regular ternary semigroups by, q-fuzzy left (right, lateral) ideals, , q-fuzzy quasi-ideals and, q-fuzzy bi-(generalized bi-) ideals.

Computational and Applied Mathematics, 2020

The main objective of the proposed work in this paper is to introduce a generalized form of rough fuzzy subsemigroups, which is rough fuzzy ternary subsemigroups (RFTSs) combining the notions of fuzziness and roughness in ternary semigroups. In RFTSs, we deal with vague and incomplete information in decision-making problems. RFTSs are characterized by lower and upper approximations using fuzzy ideals. In this research, we propose the three-dimensional k-level relation and proved that this relation is a congruence relation on a ternary semigroup. Furthermore, comparing it with the previous literature, we conclude that our proposed technique is better and effective because it deals with vague problems and there are many structures which are not handled using binary multiplication such as all the sets of negative numbers. In addition, we have proved by counterexamples that converses of many parts of many results do not hold which have negated the results proved in Q. Wang's paper. Keywords Rough ternary semigroup • Rough fuzzy ternary subsemigroup • Rough (prime) ideal • Rough fuzzy (prime) ideal Mathematics Subject Classification 20N10 • 03E72 • 18B40 • 06B10 Communicated by Marcos Eduardo Valle.

2012

In this paper, we introduce fuzzy ideals of ternary semigroups and study their related properties. Here we define fuzzy left (right, lateral) ideals of ternary semigroups and characterize regular and intra-regular ternary semigroups by using the concept of fuzzy ideals of ternary semigroups.

2013

Abstract: In this paper, we introduce the concept of (α, β)-fuzzy ideals in ternary semigroups, which is a generalization of fuzzy ideals in ternary semigroups. We investigate the related properties of ternary semigroups. The lower and upper parts of fuzzy subsets of a ternary semigroup are defined. Characterizations of regular ternary semigroups by the properties of the lower part of (∈,∈∨q)-fuzzy left (right) ideals, (∈,∈∨q)-fuzzy quasi ideals and (∈,∈∨q)-fuzzy bi-ideals are given. Key words: Ternary semigroups, (∈,∈∨q)-fuzzy ideals, (∈,∈∨q)-fuzzy left (right) ideals, (∈,∈∨q)-fuzzy quasi ideals and (∈,∈∨q)-fuzzy bi-ideals

2014

I n this paper, we introduce the definitions of anti fuzzy interior ideal and anti fuzzy characteristic interior ideals in ternary semigroups. Also we investigate some of their basic properties

Journal of the National Science Foundation of Sri Lanka, 2018

LA-semigroups are non-associative sturctures of great importance. Study of generalised roughness for fuzzy algebraic substrucures of LA-semigroups has been initiated. Many different kinds of set valued maps are needed to preserve an algebraic substrucure, while considering its lower and upper approximations. In the present paper generalised lower and upper approximations in

We consider the ternary semigroup of the fuzzy points of a ternary semigroup , and discuss the relation between some fuzzy ideals of a ternary semigroup and the subsets of