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

Front Cover
Springer Science & Business Media, Jul 31, 1999 - Computers - 203 pages
0 Reviews
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 com puter 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.
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Some Hopefully Motivating Examples
1
Background Material
7
Powers of a semiring
27
Relations with Values in a Semiring
37
Change of Base Semirings
61
Convolutions
67
Semiringvalued Subsemigroups and Submonoids
89
Semiringvalued Subgroups
113
Semiringvalued Submodules and Subspaces
141
Semiringvalued Ideals in Semirings and Rings
155
References
169
Index
189
Copyright

Common terms and phrases

Bibliographic information