Semirings

Abstract

PREFACEThe concept of a semiring generalizes that of a ring, allowing the additive sub structure to be only a semigroup instead of a group. The natural numbers provide a near at hand example of a semiring, clearly the oldest algebraic structure in which calculations have been done. Semirings occur in different mathematical fields, e. g. as ideals of a ring, as positive cones of partially ordered rings and fields, in the con text of topological considerations, and in the foundations of arithmetic, including questions raised by school education. Partially motivated by those applications, nu merous papers dealing with semirings have appeared during the last 50 years. Mean while semirings have also become of great interest as a tool in different branches of computer science.In this situation we would like to present an introduction to the algebraic the ory of semirings, including a detailed treatment of some applications in theoretical computer science. Concerning the content, we have restricted ourselves more or less to general concepts and statements of the algebraic theory of semirings and to those items of this theory which are needed for the applications mentioned above. More over, we deal with a concept of semirings that includes commutativity of addition, as is usually done for rings. So we do not consider semirings with non-commutative addition, although those also occur in the literature. Compared with the commu tative case, they are of less interest and often more difficult to handle. By the way, there are many results on semirings depending on the commutativity of addition or at least on some weakened version of it.This book is mainly written for graduate students of mathematics or computer science who are interested in understanding the algebraic theory of semirings or some of its applications. So we have tried to present our material in a corresponding way, also suitable for self-study, and have included various explaining comments, hints and cross-references. Moreover, we give complete and detailed proofs for all statements, except for some elementary and simple conclusions.On the other hand, we hope also that a reader with more experience in mathe matics or computer science will use this book as a source of quick information about the presented topics and applications of semiring theory. Therefore we have given a comprehensive bibliography and various hints to it. In this context we would like to mention the collection of literature [Gla85] by Professor K. Glazek. This, as well as our communication with him, was of great assistance to us.To help a less experienced reader, our presentation is also self-contained with respect to items needed from other mathematical domains. Only some elementary concepts of set theory are assumed to be known, but even these are mostly accom-

Cayley’s and Holland’s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices

“…Since {1} is a separating set in the i-semiring R, viewed as an R-i-semimodule, Lemma 2.2 shows that there is a semiring embedding φ : R → End(R + ), defined by (φ(r))(m) = rm, for r, m ∈ R. Thus we obtain Cayley's theorem for i-semirings, which is already known for semirings in general; see page 278 of [11]. The books [11] and [9] contain useful information on the general theory of semirings and semimodules.…”

“…The books [11] and [9] contain useful information on the general theory of semirings and semimodules. The book [10] has information about residuated semirings and semimodules.…”

Semigroup of k-bi-Ideals of a Semiring with Semilattice Additive Reduct

“…in idempotent analysis [11,12] which is now being used in theoretical physics, optimization etc. and in theoretical computer science and algorithm theory [7,10].…”

CHARACTERIZATIONS OF HEMIRINGS BY ∊,∊∨q)-FUZZY IDEALS

“…Semirings are very useful for 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 [12,13,14,22]). Hemirings, which are semirings with commutative addition and zero element appears in a natural manner in some applications to the theory of automata and formal languages (see [13,14]). …”