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Power Algebras over Semirings

With Applications in Mathematics and Computer Science

  • Book
  • © 1999

Overview

Authors:
  1. Jonathan S. Golan

    1. Department of Mathematics, University of Haifa, Haifa, Israel

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Part of the book series: Mathematics and Its Applications (MAIA, volume 488)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-x
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  2. Some (Hopefully) Motivating Examples

    • Jonathan S. Golan
    Pages 1-5

  3. Background Material

    • Jonathan S. Golan
    Pages 7-26

  4. Powers of a semiring

    • Jonathan S. Golan
    Pages 27-35

  5. Relations with Values in a Semiring

    • Jonathan S. Golan
    Pages 37-59

  6. Change of Base Semirings

    • Jonathan S. Golan
    Pages 61-66

  7. Convolutions

    • Jonathan S. Golan
    Pages 67-88

  8. Semiring-valued Subsemigroups and Submonoids

    • Jonathan S. Golan
    Pages 89-112

  9. Semiring-valued Subgroups

    • Jonathan S. Golan
    Pages 113-139

  10. Semiring-valued Submodules and Subspaces

    • Jonathan S. Golan
    Pages 141-154

  11. Semiring-valued Ideals in Semirings and Rings

    • Jonathan S. Golan
    Pages 155-167

  12. Back Matter

    Pages 169-206
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, University of Haifa, Haifa, Israel

    Jonathan S. Golan

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