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.

Redevelops the foundations of finite semigroup theory so that many major open problems fit into a single simple framework

I The q operator and Pseudovarieties of Relational Morphisms.- Foundations for Finite Semigroup Theory.- The q-operator.- The Equational Theory.- II Complexity in Finite Semigroup Theory.- The Complexity of Finite Semigroups.- Two-Sided Complexity and the Complexity of Operators.- III The Algebraic Lattice of Semigroup Pseudovarieties.- Algebraic Lattices, Continuous Lattices and Closure Operators.- The Abstract Spectral Theory of PV.- IV Quantales Idempotent Semirings Matrix Algebras and the Triangular Product.- Quantales.- The Triangular Product and Decomposition Results for Semirings.