Generalized fuzzy interior ideals in semigroups
Introduction
It is well known that l-semigroups appear (as reducts of suitable algebras) in the classical relevant logics, some non-classical logics, and multi-modal arrow logics. Note that complete l-semigroups appear in a natural manner in the theory of formal languages and programming, the theory of fuzzy sets, and the theory of automata. It is also known that some recent investigations of l-semigroups are closely connected with algebraic logic and non-classical logics. For example, algebras corresponding to general modal logics, multi-modal logics, relevance logics, and arrow logics contain l-semigroups as reducts. Some kinds of logics are closely associated with computer science. The motivation to study finite semigroups appeared in the 1950s as a result of work on linguistics and models of computation and reasoning. From such works emerged the notion of a finite automaton of which several variants can be found in the literature. A rational subset of a semigroup S is a member of the smallest set of subsets of S which contains the empty set and the singleton subsets, and is closed under union, subset product, and taking the generated subsemigroup. By observing that the empty language and one letter languages are obviously recognizable, the above shows that every rational language over a finite alphabet is recognizable. Since the inception of the notion of a fuzzy set in 1965 which laid the foundations of fuzzy set theory, the literature on fuzzy set theory and its applications has been growing rapidly amounting by now to several papers (see References). These are widely scattered over many disciplines such as artificial intelligence, computer science, control engineering, expert systems, management science, operations research, pattern recognition, robotics, and others. In [9], Yuan et al. introduced the definition of a fuzzy subgroup with thresholds which is a generalization of Rosenfeld’s fuzzy subgroup and Bhakat and Das’s fuzzy subgroup. Murali [7] proposed a definition of a fuzzy point belonging to fuzzy subset under a natural equivalence on fuzzy subset. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, which is mentioned in [8], played a vital role to generate some different types of fuzzy subgroups. It is worth pointing out that Bhakat and Das [1], [2] gave the concepts of (α,β)-fuzzy subgroups by using the “belongs to” relation (∈) and “quasi-coincident with” relation (q) between a fuzzy point and a fuzzy subgroup, and introduced the concept of an (∈, ∈ ∨ q)-fuzzy subgroup. In particular, (∈, ∈ ∨ q)-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems of other algebraic structures. As a first step in this direction, we introduce the concept of (α,β)-fuzzy interior ideals of a semigroup, and investigate related results. We discuss some fundamental aspects of (∈, ∈ ∨ q)-fuzzy interior ideals in Section 4. We show that a fuzzy set A in a semigroup S is an (∈, ∈ ∨ q)-fuzzy interior ideal of S if and only if U(A;t) is an interior ideal of S for all t ∈ (0,0.5]. This shows that (∈, ∈ ∨ q)-fuzzy interior ideals are generalizations of the existing concepts of fuzzy interior ideal. In considering the notion of an (α,β)-fuzzy interior ideal, we can consider twelve different types of such structures because there are three choices of α and four choices of β. But, in this paper, we discuss mainly (∈,∈)-type, (q,q)-type, (q,∈ ∨ q)-type, (∈ ∨ q, ∈ ∨ q)-type and (∈, ∈ ∨ q)-type.
Section snippets
Preliminaries
Let S be a semigroup. By a subsemigroup of S we mean a nonempty subset G of S such that G2 ⊆ G. A subsemigroup G of a semigroup S is called an interior ideal of S if SGS ⊆ G. A semigroup S is said to be right (resp., left) zero if xy = y (resp., xy = x) for all x,y ∈ S (see [5]).
A fuzzy set A in a set S of the formis said to be a fuzzy point with support x and value t and is denoted by xt.
For a fuzzy point xt and a fuzzy set A in a set S, Pu and Liu [8] gave meaning to the symbol
(α,β)-Fuzzy interior ideals
In what follows let S denote a semigroup, and α and β will denote any one of ∈, q, ∈ ∨ q, or ∈ ∧ q unless otherwise specified. Definition 3.1 [3], [6] A fuzzy set A in S is called a fuzzy interior ideal of S if it satisfies: (∀x,y ∈ S) (A(xy) ⩾ min{A(x),A(y)}), (∀a,x,y ∈ S) (A(xay) ⩾ A(a)).
It is well known that a fuzzy set A in S is a fuzzy interior ideal of S if and only if U(A;t)≔{x ∈ S∣A(x) ⩾ t} is an interior ideal of S for all t ∈ (0,1], for our convenience, the empty set ∅ is regarded as an interior ideal of S (see [3,
Fuzzy interior ideals of type (∈, ∈ ∨ q)
Theorem 4.1 Every (∈ ∨ q, ∈ ∨ q)-fuzzy interior ideal is an (∈, ∈ ∨ q)-fuzzy interior ideal. Proof Let A be an (∈ ∨ q, ∈ ∨ q)-fuzzy interior ideal of S. Let x,y ∈ S and t1,t2 ∈ (0,1] be such that and . Then and Since A is an (∈ ∨ q, ∈ ∨ q)-fuzzy interior ideal, it follows that Now let x,a,y ∈ S and t ∈ (0,1] be such that at ∈ A. Then at ∈ ∨ qA, and so A(xay) ⩾ A(a) ⩾ t or A(xay) + t ⩾ A(a) + t > 1. This means that (xay)t ∈ ∨ qA. Consequently, A is an (∈, ∈ ∨ q)-fuzzy interior ideal of S. □ Theorem 4.2 Every (∈,∈)-fuzzy
Conclusions
In the notion of an (α,β)-fuzzy interior ideal, we can consider twelve different types of such structures resulting from three choices of α and four choices of β. But, in this study, we have discussed mainly (∈,∈)-type, (q,q)-type, (q,∈∨q)-type, (∈ ∨ q, ∈ ∨ q)-type and (∈, ∈ ∨ q)-type.
Future research will focus on considering other types together with relations among them.
Acknowledgement
The authors are highly grateful to referees and Editor-in-Chief for their valuable comments and suggestions helpful in improving this paper.
The author Y.B. Jun was supported by Korea Research Foundation Grant (KRF-2003-005-C00013).
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