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Byline: Madad Khan, Usman Ashraf, Faisal and Abdul Sami Awan

ABSTRACT. In this paper, we have discussed some important structural properties of an ordered LA-semigroup in terms of their fuzzy interior ideals. We have shown that the set of all fuzzy interior ideals of a left regular ordered LA -semigroup with left identity forms a commutative monoid. Further, we have characterized a left regular ordered LA -semigroup by using the properties of fuzzy interior ideals, and give some equivalent statements for an ordered LA-semigroup to become a left regular ordered LA-semigroup. Finally we have given some interesting characterizations of a left (right) duo ordered LA-semigroup in terms of fuzzy interior ideals, and of a fuzzy left (right) duo ordered LA -semigroup in terms of interior ideals.

Key Words: Ordered LA-semigroup, left identity, left invertive law and fuzzy interior ideals.


The concept of fuzzy sets was first proposed by Zadeh [18] in , which has a wide range of applications in various fields such as computer engineering, artificial intelligence, control engineering, operation research, management science, robotics and many more. It gives us a tool to model the uncertainty present in phenomena that do not have sharp boundaries. Many papers on fuzzy sets have been appeared which shows the importance and its applications to set theory, algebra, real analysis, measure theory and topology etc. (see [1], [10] and [15]).

Several algebraists extended the concepts and results of algebra to the broader frame work of fuzzy set theory. Rosenfeld [15] was the first who consider the case when is a groupoid. He gave the definition of fuzzy subgroupoid and the fuzzy left (right, two-sided) ideal of and justified these definitions by showing that a subset of a groupoid is a subgroupoid or a left (right, two-sided) ideal of if the characteristic function of , that is


is a fuzzy subgroupoid or a fuzzy left (right, two-sided) ideal of .Kuroki and Mordeson have widely explored fuzzy semigroups in [10] and [11].

Fuzzy algebra is going popular day by day due to wide applications of fuzzification in almost every field. Our aim in this paper is to develop some characterizations for a new non-associative algebraic structure known as a left almost semigroup which is the generalization of a commutative semigroup (see [4]). A left almost semigroup is an algebraic structure mid way between a groupoid and a commutative semigroup. A left almost semigroup has wide range of applications in theory of flocks (see [3]).

A left almost semigroup, abbreviated as an -semigroup is a groupoid whose elements satisfy the left invertive law , The concept of this algebraic structure was first introduced by Kazim and Naseeruddin in 1972 [4]. In an -semigroup, the medial law [4] holds . An -semigroup may or may not contain a left identity. The left identity of an -semigroup allow us to introduce the inverses of elements in an -semigroup. If an -semigroup contains a left identity, then it is unique [12]. In an -semigroup with left identity, the paramedial law holds By using medial law with left identity, we get , Several examples and interesting properties of -semigroups can be found in [12] and [16].

An -semigroup is a non-associative and non-commutative algebraic structure mid way between a groupoid and a commutative semigroup. This structure is closely related with a commutative semigroup, because if an -semigroup contains a right identity, then it becomes a commutative semigroup [12]. The connection of a commutative inverse semigroup with an -semigroup has been given in [13] as, a commutative inverse semigroup , becomes an -semigroup , under . An -semigroup with left identity becomes a semigroup under the binary operation " " defined as, if , , there exists such that [16]. An -semigroup is the generalization of a semigroup theory [12] and has vast applications in collaboration with semigroup like other branches of mathematics. The connection of -semigroups with the vector spaces over finite fields has been investigated in [7].

From the above discussion, we see that -semigroups have very closed links with semigroups and vector spaces which shows the importance and applications of this non-associative algebraic structure.

An ordered -semigroup ( - -semigroup) [9] is a structure in which the following conditions hold. is an -semigroup is a poset.

Assume that is an ordered -semigroup, and let denotes the set of all fuzzy subsets of then is an ordered -semigroup and satisfies all the basic laws of an ordered -semigroup [9].

2. Preliminaries

In this section, we have given some basic definitions which are necessary for the subsequent sections. Throughout in this paper will be considered as an ordered -semigroup unless otherwise specified.



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Author:Khan, Madad; Ashraf, Usman; Awan, Abdul Sami
Publication:Science International
Article Type:Report
Geographic Code:9PAKI
Date:Jun 30, 2012

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