On fuzzy h-ideals in hemirings
Introduction
Semirings play an important role in studying matrices and determinants. Many aspects of the theory of matrices and determinants over semirings have been studied by Beasley and Pullman [3], [4], Ghosh [9], and others. Although ideals in semirings are useful for many purposes, they do not in general coincide with the usual ring ideals if S is a ring and, for this reason, their use is somewhat limited in trying to obtain analogues of ring theorems for semirings. Indeed, many results in rings apparently have no analogues in semirings using only ideals. Henriksen [11] defined a more restricted class of ideals in semirings, which is called k-ideals, with the property that if the semiring S is a ring then a complex in S is a k-ideal if and only if it is a ring ideal. A still more restricted class of ideals in hemirings has been given by Iizuka [12]. However, a definition of ideal in any additively commutative semiring S can be given which coincides with Iizuka's definition provided S is a hemiring, and it is called h-ideal. La Torre [20] investigated h-ideals and k-ideals in hemirings in an effort to obtain analogues of familiar ring theorems. The concept of a fuzzy set, introduced in Zadeh [21], was applied to generalize some of the basic concepts of algebra. Several authors have discussed a fuzzy theory in semirings (see [1], [2], [6], [7], [8], [10], [13], [14], [15], [16], [18], [19]). In this paper, we consider the fuzzy setting of h-ideals in hemirings. Using a collection of left h-ideals of a hemiring S, we establish fuzzy left h-ideals of S. We give a characterization of a finite valued fuzzy left h-ideal in a hemiring S, and show that if S is an h-Noetherian hemiring, then every fuzzy left h-ideal of S is finite valued. We prove that if μ and ν are fuzzy left h-ideals of a hemiring S, then μ×ν is a fuzzy left h-ideal of S×S. Conversely, we show that if μ×ν is a fuzzy left h-ideal of S×S, then either μ or ν is a fuzzy left h-ideal of S. We prove that a fuzzy set ν in a hemiring S is a fuzzy left h-ideal of S if and only if the strongest fuzzy relation μν on S is a fuzzy left h-ideal of S×S.
Section snippets
Preliminaries
A semiring S is a system consisting of a nonempty set S together with two binary operations on S called addition and multiplication (denoted in the usual manner) such that
- •
S together with addition is a semigroup,
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S together with multiplication is a semigroup, and
- •
a(b+c)=ab+ac and (a+b)c=ac+bc for all a,b,c∈S.
A semiring S is said to be additively commutative if a+b=b+a for all a,b∈S. A zero element of a semiring S is an element 0 such that 0·x=x·0=0 and 0+x=x+0=x for all x∈S. By a hemiring we mean
Fuzzy h-ideals in hemirings
Definition 3.1 A left h-ideal of a hemiring S is defined to be a left ideal A of S such that Right h-ideals are defined similarly. Note that every left (resp. right) h-ideal is a left (resp. right) k-ideal, but the converse may not be true (see [20]). Definition 3.2 A fuzzy left h-ideal of a hemiring S is defined to be a fuzzy left ideal μ of S such that[20]
Fuzzy right h-ideals are defined similarly. Theorem 3.3 Let A be a nonempty subset of a hemiring S. Let μ be a
Cartesian product of fuzzy left h-ideals
Definition 4.1 A fuzzy relation on any set S is a fuzzy set Definition 4.2 If μ is a fuzzy relation on a set S and ν is a fuzzy set in S, then μ is a fuzzy relation on ν if Definition 4.3 Let μ and ν be fuzzy sets in a set S. The Cartesian product of μ and ν is defined by Lemma 4.4 Let μ and ν be fuzzy sets in a set S. Then μ×ν is a fuzzy relation on S, U(μ×ν;t)=U(μ;t)×U(ν;t) for all t∈[0,1].[5]
[5]
[5]
[5]
Definition 4.5 [5]
If ν is a fuzzy set in a set S, the strongest fuzzy relation on S that is a fuzzy relation
Acknowledgements
The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper. The first and third authors were supported by grant R01-2001-000-00004-0(2002) from the Korea Science and Engineering Foundation.
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