Ebook: Semirings and their Applications

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. - Nikolai Ivanovich Lobatchevsky This book is an extensively-revised and expanded version of "The Theory of Semirings, with Applicationsin Mathematics and Theoretical Computer Science" [Golan, 1992], first published by Longman. When that book went out of print, it became clear - in light of the significant advances in semiring theory over the past years and its new important applications in such areas as idempotent analysis and the theory of discrete-event dynamical systems - that a second edition incorporating minor changes would not be sufficient and that a major revision of the book was in order. Therefore, though the structure of the first «dition was preserved, the text was extensively rewritten and substantially expanded. In particular, references to many interesting and applications of semiring theory, developed in the past few years, had to be added. Unfortunately, I find that it is best not to go into these applications in detail, for that would entail long digressions into various domains of pure and applied mathematics which would only detract from the unity of the volume and increase its length considerably. However, I have tried to provide an extensive collection of examples to arouse the reader's interest in applications, as well as sufficient citations to allow the interested reader to locate them. For the reader's convenience, an index to these citations is given at the end of the book .

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This work is an updated and considerably expanded version of the author's book The Theory of Semirings, with Applications toMathematics and Theoretical Science, which has been recognized as the definitive reference work in this area. This edition includes many of the new results in this area, as well as further applications of semiring theory in such areas as idempotent analysis, discrete dynamical systems, formal language theory, fuzzy set theory, optimization etc. The book contains an extensive bibliography and a large number of examples.

Audience: This book is aimed both at mathematicians and at researchers in applied mathematics and theoretical computer science. It is also suitable for use as a graduate-level textbook.

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This work is an updated and considerably expanded version of the author's book The Theory of Semirings, with Applications toMathematics and Theoretical Science, which has been recognized as the definitive reference work in this area. This edition includes many of the new results in this area, as well as further applications of semiring theory in such areas as idempotent analysis, discrete dynamical systems, formal language theory, fuzzy set theory, optimization etc. The book contains an extensive bibliography and a large number of examples.

Audience: This book is aimed both at mathematicians and at researchers in applied mathematics and theoretical computer science. It is also suitable for use as a graduate-level textbook.

Content:
Front Matter....Pages i-xi
Hemirings and Semirings: Definitions and Examples....Pages 1-18
Sets and Relations with Values in a Semiring....Pages 19-25
Building New Semirings from Old....Pages 27-42
Some Conditions on Semirings....Pages 43-57
Complemented Elements in Semirings....Pages 59-63
Ideals in Semirings....Pages 65-83
Prime and Semiprime Ideals in Semirings....Pages 85-94
Factor Semirings....Pages 95-103
Morphisms of Semirings....Pages 105-120
Kernels of Morphisms....Pages 121-127
Semirings of Fractions....Pages 129-134
Euclidean Semirings....Pages 135-141
Additively-Regular Semirings....Pages 143-147
Semimodules over Semirings....Pages 149-161
Factor Semimodules....Pages 163-179
Some Constructions for Semimodules....Pages 181-190
Free, Projective, and Injective Semimodules....Pages 191-202
Localization of Semimodules....Pages 203-210
Linear Algebra over a Semiring....Pages 211-221
Partially-Ordered Semirings....Pages 223-237
Lattice-Ordered Semirings....Pages 239-246
Complete Semirings....Pages 247-258
Complete Semimodules....Pages 259-265
CLO-Semirings....Pages 267-283
Fixed Points of Affine Maps....Pages 285-306
Back Matter....Pages 307-381

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This work is an updated and considerably expanded version of the author's book The Theory of Semirings, with Applications toMathematics and Theoretical Science, which has been recognized as the definitive reference work in this area. This edition includes many of the new results in this area, as well as further applications of semiring theory in such areas as idempotent analysis, discrete dynamical systems, formal language theory, fuzzy set theory, optimization etc. The book contains an extensive bibliography and a large number of examples.

Audience: This book is aimed both at mathematicians and at researchers in applied mathematics and theoretical computer science. It is also suitable for use as a graduate-level textbook.

Content:
Front Matter....Pages i-xi
Hemirings and Semirings: Definitions and Examples....Pages 1-18
Sets and Relations with Values in a Semiring....Pages 19-25
Building New Semirings from Old....Pages 27-42
Some Conditions on Semirings....Pages 43-57
Complemented Elements in Semirings....Pages 59-63
Ideals in Semirings....Pages 65-83
Prime and Semiprime Ideals in Semirings....Pages 85-94
Factor Semirings....Pages 95-103
Morphisms of Semirings....Pages 105-120
Kernels of Morphisms....Pages 121-127
Semirings of Fractions....Pages 129-134
Euclidean Semirings....Pages 135-141
Additively-Regular Semirings....Pages 143-147
Semimodules over Semirings....Pages 149-161
Factor Semimodules....Pages 163-179
Some Constructions for Semimodules....Pages 181-190
Free, Projective, and Injective Semimodules....Pages 191-202
Localization of Semimodules....Pages 203-210
Linear Algebra over a Semiring....Pages 211-221
Partially-Ordered Semirings....Pages 223-237
Lattice-Ordered Semirings....Pages 239-246
Complete Semirings....Pages 247-258
Complete Semimodules....Pages 259-265
CLO-Semirings....Pages 267-283
Fixed Points of Affine Maps....Pages 285-306
Back Matter....Pages 307-381
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