Fundamentals Free inverse semigroups and the theorems of McAlister Biordered sets Zigzags and their applications Semigroup diagrams and word problems Combinatorial aspects of transformation semigroups References Glossary of notation Index.

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.

Techniques of semigroup theory

The q-theory of Finite Semigroups

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background

Classical finite transformation semigroups

https://u.cs.biu.ac.il/~margolis/Representation%20Theory%20Seminar/Monoid%20Representations%20and%20Applications/Solomon%20Rook%20presentation.pdf

Representations of the rook monoid

Generators and relations for partition monoids and algebras

It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n − 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n − 1 while the analogous result is true for the semigroup of all n × n matrices over a field. In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ϵ . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow. In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ϵ is an idempotent of rank n − 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup. The fourth section deals with a special ca...

Techniques of semigroup theory

Citations

The q-theory of Finite Semigroups

Classical finite transformation semigroups

Representations of the rook monoid

Generators and relations for partition monoids and algebras

Semigroups generated by a group and an idempotent

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