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The purpose of this book is to present an up to date account of fuzzy ideals of a semiring. The book concentrates on theoretical aspects and consists of eleven chapters including three invited chapters. Among the invited chapters, two are devoted to applications of Semirings to automata theory, and one deals with some generalizations of Semirings. This volume may serve as a useful hand book for graduate students and researchers in the areas of Mathematics and Theoretical Computer Science.



Part I


Fundamental Concepts

This introductory chapter comprises 6 sections. In section 1, we give a general introduction to the algebraic structure of semirings. Section 2 contains basic definitions, examples, and important applications of semirings. In section 3, we assemble preliminary definitions and results, and section 4 provides a summary of algebraic preliminaries related to the structure of semirings. In section 5, we present the concept of fuzzy sets which was introduced by Lotfi A. Zadeh in 1965, collect basic definitions, examples, and applications. In the final section 6 of this chapter, we give a brief review of various Fuzzy algebraic structures.
Javed Ahsan, John N. Mordeson, Mohammad Shabir

Fuzzy Ideals of Semirings

Section 1 of this chapter begins with the definitions of sums and products of fuzzy sets and fuzzy ideals of a semiring together with some basic results related to these definitions. It is shown that the sum of fuzzy left (right) ideals, product of left (right) ideals is a fuzzy left (right) ideal. It is further shown that a fuzzy subset of a semiring R is a fuzzy left (right) ideal of R if and only if each nonempty level subset (as defined in 2.1) is a left (right) ideal of R. In section 2, we characterize Regular semirings in terms of their fuzzy left and fuzzy right ideals. Characterizations of weakly regular semirings in terms of fuzzy left (right) ideals are given in section 3. Finally in section 4, we collect various characterizations of fully idempotent semirings in terms of their fuzzy ideals. Here we also give characterizations of fully idempotent semirings in terms of their fuzzy prime ideals (cf. Theorem 2.13).
Javed Ahsan, John N. Mordeson, Mohammad Shabir

Fuzzy Subsemimodules over Semirings

It is well-known that modules are a generalization of vector spaces of linear algebra in which the ”scalars” are allowed to be from an arbitrary ring, rather than a field. This rather modest weakening of the axioms is quite far reaching, including, for example, the theory of rings and ideals and the theory of abelian groups as special cases.
Javed Ahsan, John N. Mordeson, Mohammad Shabir

Fuzzy k-Ideals of Semirings

This chapter consists of eight sections. In section 1 we prove that the k-sum and k-product of fuzzy k-ideals of a semiring is a fuzzy k-ideal. section 2 is devoted to characterizing k-regular semirings in terms of fuzzy left (right) k-ideals. Section 3 contains various characterizations of right k-weakly regular semirings by fuzzy right k-ideals. In section 4, we define fuzzy prime and semiprime right k-ideals and it is shown that R is a right k-weakly regular semiring if and only if each fuzzy right kideal of R is semiprime. In section 5, we characterize semirings in which each fuzzy k-ideal is idempotent and section 6 presents some results on prime and semiprime fuzzy k-ideals. We then study ”k-semirings” and present some results related to these semirings. Finally, Section 8 is devoted to a study of fuzzy congruences of semirings.
Javed Ahsan, John N. Mordeson, Mohammad Shabir

Fuzzy Quasi-ideals and Fuzzy Bi-ideals in Semirings

The object of this chapter is to study fuzzy quasi-ideals and fuzzy bi-ideals and Section 1 provides a study of these ideals. Section 2 presents various characterizations of regular semirings involving these fuzzy ideals. In Section 3, we examine and characterize regular and intra regular semirings in this context. Section 4 provides a study of fuzzy k-quasi-ideals and fuzzy k-bi-ideals of semirings and Section 5 contains characterizations of k-regular semirings by the ideals studied in Section 4. Section 6 contains characterizations of k-intra regular semirings by the ideals examined in the preceding section.
Throughout this chapter R will denote a semiring with zero.
Javed Ahsan, John N. Mordeson, Mohammad Shabir

(∈, ∈ Vq)-Fuzzy Ideals in Semirings

Following Bhakat and Das [21], Dudek et. al. [47], Ma and Zhan [106] defined (∈, ∈ Vq)-fuzzy ideals in semirings. In this chapter we have studied properties of these fuzzy ideals. In Section 1, we look at (α,β)-fuzzy ideals, and show that a fuzzy left (right) ideal of a semiring is an (∈, ∈)-fuzzy left (right) ideal. Section 2 provides a characterization of (∈, ∈ Vq)-fuzzy left (right) ideals, quasi and bi-ideals of semirings. Section 3 presents characterizations of regular semirings involving these ideals and Section 4 contains characterizations of regular and intra regular semirings by these ideals. Section 5 presents a study of (∈, ∈ Vq)-fuzzy k-ideals, k-quasi-ideals and k-bi-ideals of semirings. A study of k-regular and k-intra regular semirings in this context is separately made in Sections 6 and 7.
Throughout this chapter R is a semiring with zero element.
Javed Ahsan, John N. Mordeson, Mohammad Shabir

( $\overline\in$ , $\overline\in$ Vq̄)-Fuzzy Ideals in Semirings

This chapter is devoted to a study of (\(\overline\in\), \(\overline\in\) Vq̄)-fuzzy ideals, fuzzy quasi-ideals and bi-ideals of semirings on the pattern of Chapter 6.
Javed Ahsan, John N. Mordeson, Mohammad Shabir

Fuzzy Ideals with Thresholds

Modelled on the previous two chapters, we make a similar study of fuzzy ideals, fuzzy quasi-ideals and fuzzy bi-ideals with thresholds in this chapter.
Javed Ahsan, John N. Mordeson, Mohammad Shabir

Invited Chapters


On Fuzzy LD-Bigroupoids

In this paper, we define a generalization of what it means to be a semiring, i.e., the class of LD-bigroupoids with companion classes RD-bigroupoids and D-bigroupoids. After development of several basic ideas we consider the fuzzified versions of these algebraic systems and we investigate how theideas developed for LD-bigroupoids carry over into the realm of fuzzy LD-bigroupoids, yielding a generalization of the theory of fuzzy semirings. The results obtained demonstrate that it is quite possible to take these ideas much further as we expect with happen in the future.
2000 Mathematics Subject Classification. 03E72, 16Y60.
Hee Sik Kim, J. Neggers

Semiring Parsing

Syntactic parsing is an important task in natural language processing (NLP). In this chapter an application of semiring theory in parsing (a.k.a.”semiring parsing”) will be introduced. A semiring parsing framework is proposed and studied in [6].
Yudong Liu

Coverings and Decompositions of Semiring-Weighted Finite Transition Systems

We consider weighted finite transition systems (WTS) with weights from naturally ordered semirings. Such semirings comprise the natural numbers with ordinary addition and multiplication as well as distributive lattices and the max -plus-semiring. For these systems we explore the concepts of covering and cascade product. We show a cascade decomposition result for such WTS using special partitions of the state set of the system. This extends a classical result of automata theory to the weighted setting.
Manfred Droste, Ingmar Meinecke, Branimir Šešelja, Andreja Tepavčević


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