Elsevier

Information Sciences

Volume 178, Issue 17, 1 September 2008, Pages 3451-3464
Information Sciences

The characterizations of h-hemiregular hemirings and h-intra-hemiregular hemirings

https://doi.org/10.1016/j.ins.2008.04.002Get rights and content

Abstract

In this paper, the concepts of fuzzy h-bi-ideals and fuzzy h-quasi-ideals of a hemiring are introduced, and related properties are investigated. The notion of h-intra-hemiregularity of a hemiring, which is a generalization of the notion of intra-regularity of a ring, is provided. Some characterizations of h-hemiregular hemirings, h-intra-hemiregular hemirings and hemirings that are both h-hemiregular and h-intra-hemiregular are derived in terms of fuzzy left, fuzzy right h-ideals, fuzzy h-bi-ideals and fuzzy h-quasi-ideals.

Introduction

As a generalization of rings, semirings have been found useful for solving problems in different areas of applied mathematics and information sciences, since the structure of a semiring provides an algebraic framework for modelling and studying the key factors in these applied areas. They play an important role in studying optimization theory, graph theory, theory of discrete event dynamical systems, matrices, determinants, generalized fuzzy computation, automata theory, formal language theory, coding theory, analysis of computer programs, and so on (see [3], [4], [6], [8], [10], [23], [24] for details). Ideals of semirings play a central role in the structure theory and are useful for many purposes. However, they do not in general coincide with the usual ring ideals 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. In order to overcome this deficiency, Henriksen [11] defined a more restricted class of ideals in semirings, which is called the class of 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]. According to Iizuka’s definition, an ideal in any additively commutative semiring S can be given which coincides with a ring ideal provided S is a hemiring, and it is called h-ideal. The properties of h-ideals and k-ideals of hemirings were thoroughly investigated by La Torre [19] and by using the h-ideals and k-ideals, La Torre established some analogous ring theorems for hemirings.

The theory of fuzzy sets, proposed by Zadeh [25] in 1965, has provided a useful mathematical tool for describing the behavior of systems that are too complex or illdefined to admit precise mathematical analysis by classical methods and tools. Extensive applications of fuzzy set theory have been found in various fields such as artificial intelligence, computer science, control engineering, expert systems, management science, operations research, pattern recognition, robotics, and others. It soon invoked a natural question concerning a possible connection between fuzzy sets and algebraic systems. The study of the fuzzy algebraic structures has started in the pioneering paper of Rosenfeld [22] in 1971. Rosenfeld introduced the notion of fuzzy groups and showed that many results in groups can be extended in an elementary manner to develop the theory of fuzzy group. Since then the literature of various fuzzy algebraic concepts has been growing very rapidly. For example, Kuroki [20], [21] introduced and studied the concept of fuzzy ideals of a semigroup. Subsequently, many authors fuzzified certain standard concepts and results on rings and modules. In [15], [16], Kehayopulu and Tsingelis applied the concept of fuzzy sets to the theory of ordered semigroups and characterized the regular ordered semigroups in terms of fuzzy ideals. The relationships between the fuzzy sets and semirings (hemirings) have been considered by Dutta, Baik, Ghosh, Jun, Kim, Zhan and others. The reader is refereed to [1], [2], [5], [6], [7], [9], [13], [14], [17], [26].

Recently, Zhan et al. [27] introduced the concept of h-hemiregularity of a hemiring and gave a characterization of h-hemiregular hemirings in terms of fuzzy right and fuzzy left h-ideals. As a continuation of the paper [27], we consider the characterizations of h-hemiregular hemirings and h-intra-hemiregular hemirings. We introduce the concepts of fuzzy h-bi-ideals and fuzzy h-quasi-ideals of a hemiring, and give some of their properties. We provide the notion of h-intra-hemiregularity of a hemiring as a generalization of the notion of intra-regularity of a ring. Further, we investigate the characterizations of h-hemiregular hemirings, h-intra-hemiregular hemirings and hemirings that are both h-hemiregular and h-intra-hemiregular in terms of fuzzy left, fuzzy right h-ideals, fuzzy h-bi-ideals and fuzzy h-quasi-ideals.

Section snippets

Preliminaries

A semiring is an algebraic system (S,+,·) consisting of a non-empty set S together with two binary operations on S called addition and multiplication (denoted in the usual manner) such that (S,+) and (S,·) are semigroups and the following distributive lawsa(b+c)=ab+bcand(a+b)c=ac+bcare satisfied for all a,b,cS.

By zero of a semiring (S,+,·) we mean an element 0S such that 0·x=x·0=0 and 0+x=x+0=x for all xS. A semiring with zero and a commutative semigroup (S,+) is called a hemiring. For the

Fuzzy h-ideals in hemirings

It is well known that ideal theory plays a fundamental role in the development of hemirings. In [14], Jun et al. introduced the concepts of fuzzy left and fuzzy right h-ideals of a hemiring. In this section, we define the notions of fuzzy h-bi-ideals and fuzzy h-quasi-ideals of a hemiring, and investigate some of their properties.

Definition 3.1

[14]

A fuzzy subset μ in a hemiring S is called a fuzzy left h-ideal if for all x,y,z,a,bS we have

  • (i)

    μ(x+y)min{μ(x),μ(y)},

  • (ii)

    μ(xy)μ(y),

  • (iii)

    x+a+z=b+zμ(x)min{μ(a),μ(b)}.

Fuzzy

h-Hemiregular hemirings

The concept of h-hemiregularity of a hemiring was first introduced by Zhan et al. [27] as a generalization of the concept of regularity of a ring. In this section, we concentrate our study on the characterizations of h-hemiregular hemirings. We start by formulating the following definition.

Definition 4.1

[27]

A hemiring S is said to be h-hemiregular if for each xS, there exist a,a,zS such that x+xax+z=xax+z.

Lemma 4.2

[27]

A hemiring S is h-hemiregular if and only if for any right h-ideal R and any left h-ideal L of S we have

h-Intra-hemiregular hemirings

In this section, we introduce the concept of h-intra-hemiregularity of a hemiring, and investigate the characterizations of h-intra-hemiregular hemirings and hemirings that are both h-hemiregular and h-intra-hemiregular.

We first give the concept of h-intra-hemiregularity of a hemiring as follows.

Definition 5.1

A hemiring S is said to be h-intra-hemiregular if for each xS, there exist ai,ai,bj,bj,zS such that x+i=1maix2ai+z=j=1nbjx2bj+z. Equivalent definitions: (1) xSx2S¯xS, (2) ASA2S¯AS.

It is

Conclusions

Since Zadeh proposed the notion of fuzzy sets, his ideas have been applied to various fields. In the paper, we applied these ideas to hemirings. We introduced the concepts of fuzzy h-bi-ideals and fuzzy h-quasi-ideals of a hemiring, and gave some of their properties. We provided the notion of h-intra-hemiregularity of a hemiring as a generalization of the notion of intra-regularity of a ring. We also investigated the characterizations of h-hemiregular hemirings, h-intra-hemiregular hemirings

Acknowledgements

We express our warmest thanks to Professor Witold Pedrycz, Editor-in-Chief, for editing, communicating the paper, and his useful suggestions. We also express our warmest thanks to the referees for their interest in our work and their value time to read the manuscript very carefully and their valuable comments for improving the paper. This research was supported by National Natural Science Foundation of China (60774049) and Major State Basic Research Development Program of China (2002CB312200).

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