Smarandache-Alpha Fuzzy Normal Subsemigroups in Smarandache-Alpha Fuzzy Semigroups
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RESEARCH ARTICLE
Smarandache-Alpha Fuzzy Normal Subsemigroups in Smarandache-Alpha Fuzzy Semigroups R. Gowri1 • T. Rajeswari2
Received: 3 March 2016 / Revised: 20 April 2016 / Accepted: 24 May 2019 Ó The National Academy of Sciences, India 2019
Abstract The notion of S-a fuzzy cosets with representative xt and S-a fuzzy normal subsemigroups are introduced, and some equivalent conditions are given. It is also proved that the set of all S-a fuzzy cosets will form a semigroup under a suitable binary operation and its structure properties are determined. A necessary and sufficient condition for an S-a fuzzy semigroup to be S-a fuzzy normal is also proved. Keywords S-semigroup Fuzzy group S-fuzzy group a-fuzzy set a-fuzzy group S-a fuzzy semigroup Mathematics Subject Classification 03E72 08A72 20N25
Significance statement
It is proved that A/B is completely regular, where A and B are S-a fuzzy semigroups of an S-semigroup and it is illustrated that A/B need not be a group by a suitable example. & T. Rajeswari [email protected] R. Gowri [email protected] 1
Department of Mathematics, Government College for Women(Autonomous), Kumbakonam, India
2
Department of Mathematics, Idhaya College for Women, Kumbakonam, India
1 Introduction and Preliminaries The concept of fuzzy set was introduced by Zadeh [1]. A Fuzzy subgroup of a group was defined by Rosenfeld [2]. Fuzzy group was redefined by Anthony and Sherwood [3]. Liu [4] initiated the study of Fuzzy invariant subgroups and fuzzy ideals in 1982. A characterization of fuzzy subgroups was obtained by Anthony and Sherwood [5]. Mashour et al. [6] defined normal fuzzy subgroups in 1990. Malik et al. [7] introduced the notion of fuzzy cosets with representative xt and fuzzy normal of B in A, where A and B are fuzzy subgroups of a group G such that B A. Fuzzy algebra was studied by Rajeshkumar [8]. Vasantha Kandasamy [9] studied about Smarandache fuzzy semigroups. Quotient groups induced by fuzzy subgroups were analyzed by Liu [10]. Massa’deh [11] introduced fuzzy subgroups with operators. Sharma [12] introduced the concept of a-fuzzy set, a-fuzzy group, a-fuzzy coset, a-fuzzy normal subsemigroups and obtained their properties. Gowri and Rajeswari [13] introduced the idea of S-a fuzzy semigroup, S-a fuzzy left, right cosets and S-a fuzzy normal subsemigroup and analyzed their properties. In this paper, the concepts of S-a fuzzy cosets with representative xt and S-a fuzzy normal subsemigroups are introduced. These ideas differ from those in [13]. It is also proved that the set of all S-a fuzzy cosets will form a semigroup under a suitable binary operation and its structure properties are determined. Throughout this paper, a will always denote a member of [0, 1]. Definition 1.1 Let X be a nonempty set. A fuzzy subset A of X is a function A : X ! ½0; 1. Definition 1.2 A fuzzy subset A of a group G is called a fuzzy subgroup of G if
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(i) AðxyÞ minfAðxÞ; AðyÞg (ii) Aðx1 Þ ¼ AðxÞ, for all x; y 2 G.
2 S-a fuzzy coset
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