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

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.

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This research monograph investigates sets of the form R^A where R is a semiring, and A is a set with a certain structure. Such constructs generalise fuzzy and toll algebraic structures, which in recent years have shown themselves of importance in engineering and computer science, as well as in new mathematical disciplines like idempotent analysis. This volume puts much of the ad hoc work of the past decade on a firm mathematical foundation by creating a unified approach which has important implications in such diverse areas as formal language theory, discrete dynamical systems, models of optimal control, and many others.

This book also seeks to address the fundamental question posed by H?hle and Sostak as to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory.

Since the development of a new mathematical theory is basically inductive, this book begins with a large number of instances which appear in various mathematical contexts, and then develops a general abstract framework, with many examples which will encourage further study in greater depth.

Audience: This work will be of interest to mathematicians whose work involves associative rings and algebras, ordered algebraic structures, optimal control, graph theory, fuzzy sets and fuzzy algebraic structures.

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This research monograph investigates sets of the form R^A where R is a semiring, and A is a set with a certain structure. Such constructs generalise fuzzy and toll algebraic structures, which in recent years have shown themselves of importance in engineering and computer science, as well as in new mathematical disciplines like idempotent analysis. This volume puts much of the ad hoc work of the past decade on a firm mathematical foundation by creating a unified approach which has important implications in such diverse areas as formal language theory, discrete dynamical systems, models of optimal control, and many others.

This book also seeks to address the fundamental question posed by H?hle and Sostak as to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory.

Since the development of a new mathematical theory is basically inductive, this book begins with a large number of instances which appear in various mathematical contexts, and then develops a general abstract framework, with many examples which will encourage further study in greater depth.

Audience: This work will be of interest to mathematicians whose work involves associative rings and algebras, ordered algebraic structures, optimal control, graph theory, fuzzy sets and fuzzy algebraic structures.

Content:
Front Matter....Pages i-x
Some (Hopefully) Motivating Examples....Pages 1-5
Background Material....Pages 7-26
Powers of a semiring....Pages 27-35
Relations with Values in a Semiring....Pages 37-59
Change of Base Semirings....Pages 61-66
Convolutions....Pages 67-88
Semiring-valued Subsemigroups and Submonoids....Pages 89-112
Semiring-valued Subgroups....Pages 113-139
Semiring-valued Submodules and Subspaces....Pages 141-154
Semiring-valued Ideals in Semirings and Rings....Pages 155-167
Back Matter....Pages 169-206

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This research monograph investigates sets of the form R^A where R is a semiring, and A is a set with a certain structure. Such constructs generalise fuzzy and toll algebraic structures, which in recent years have shown themselves of importance in engineering and computer science, as well as in new mathematical disciplines like idempotent analysis. This volume puts much of the ad hoc work of the past decade on a firm mathematical foundation by creating a unified approach which has important implications in such diverse areas as formal language theory, discrete dynamical systems, models of optimal control, and many others.

This book also seeks to address the fundamental question posed by H?hle and Sostak as to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory.

Since the development of a new mathematical theory is basically inductive, this book begins with a large number of instances which appear in various mathematical contexts, and then develops a general abstract framework, with many examples which will encourage further study in greater depth.

Audience: This work will be of interest to mathematicians whose work involves associative rings and algebras, ordered algebraic structures, optimal control, graph theory, fuzzy sets and fuzzy algebraic structures.

Content:
Front Matter....Pages i-x
Some (Hopefully) Motivating Examples....Pages 1-5
Background Material....Pages 7-26
Powers of a semiring....Pages 27-35
Relations with Values in a Semiring....Pages 37-59
Change of Base Semirings....Pages 61-66
Convolutions....Pages 67-88
Semiring-valued Subsemigroups and Submonoids....Pages 89-112
Semiring-valued Subgroups....Pages 113-139
Semiring-valued Submodules and Subspaces....Pages 141-154
Semiring-valued Ideals in Semirings and Rings....Pages 155-167
Back Matter....Pages 169-206
....