This comprehensive, encyclopedic text in four parts aims to give the reader — from the graduate student to the researcher/practitioner — a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, thereby updating and modernizing the semigroup theory literature.
Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner – with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research.
Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises.
Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index.
The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.
-Preface.- List of Tables.-List of Figures.-Introduction.-Part I. The q-operator and Pseudovarieties of Relational Morphisms.-1. Foundations for Finite Semigroup Theory.-2. The q-operator - 3.The Equational Theory -Part II. Complexity in Finite Semigroup Theory. -4. The Complexity of Finite Semigroups. -5. Two-Sided Complexity and the Complexity of Operators.-Part III. The Algebraic Lattice of Semigroup Pseudovarieties.-6. Algebraic Lattices, Continuous Lattices and Closure Operators.-7. The Abstract Spectral Theory of PV.-Part IV. Quantales, Indempotent Semirings, Matrix Algebras and the Triangular Product-8. Quantales.-9. The Triangular Product and Decomposition Results for Semirings.-A. The Green-Rees Local Structure Theory.-B. Tables on Preservation of Sups and Infs.- List of Problems.- References.- Table of Pseudovarieties.-Table of Operators and Products.-Index of Notation.-Author Index.-Index.
Redevelops the foundations of finite semigroup theory so that many major open problems fit into a single simple framework
A new theory is developed. It provides a unifying approach to finite semigroup theory via quantization
Many important techniques and results are clearly exposited in book form for the first time, thereby updating and modernizing the semigroup theory literature and placing all the most important results into context
Introduces Spectral Theory into Finite Semigroup Theory
Over 70 research problems are presented, many of which can serve as the basis for a doctoral thesis
Encyclopedic in that the reader can find all the results
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