Power Algebras over Semirings: With Applications in Mathematics and Computer Science / Edition 1

by Jonathan S. Golan

Overview

This monograph is a continuation of several themes presented in my previous books [146, 149]. In those volumes, I was concerned primarily with the properties of semirings. Here, the objects of investigation are sets of the form RA, where R is a semiring and A is a set having a certain structure. The problem is one of translating that structure to RA in some "natural" way. As such, it tries to find a unified way of dealing with diverse topics in mathematics and theoretical computer science as formal language theory, the theory of fuzzy algebraic structures, models of optimal control, and many others. Another special case is the creation of "idempotent analysis" and similar work in optimization theory. Unlike the case of the previous work, which rested on a fairly established mathematical foundation, the approach here is much more tentative and docimastic. This is an introduction to, not a definitative presentation of, an area of mathematics still very much in the making. The basic philosphical problem lurking in the background is one stated suc­ cinctly by Hahle and Sostak [185]: ". . . to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory?" The conflicting definitions proposed by various researchers in search of a resolution to this conundrum show just how difficult this problem is to see in a proper light.

Product Details

ISBN-13:	9789048152704
Publisher:	Springer Netherlands
Publication date:	12/08/2010
Series:	Mathematics and Its Applications , #488
Edition description:	Softcover reprint of hardcover 1st ed. 1999
Pages:	206
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

Some (hopefully) motivating examples.- 0: Background material.- 1: Powers of a semiring.- 2: Relations with values in a semiring.- 3: Change of base semirings.- 4: Convolutions.- 5: Semiring-valued subsemigroups and submonoids.- 6: Semiring-valued groups.- 7: Semiring-valued submodules and subspaces.- 8: Semiring-valued ideals in semirings and rings.- References.