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.

The q-theory of Finite Semigroups

Left restriction semigroups are a class of semigroups which generalise inverse semigroups and which emerge very naturally from the study of partial transformations of a set.  Consequently, they have arisen in a variety of different contexts, under a range of names.  One of the various guises under which left restriction semigroups have appeared is that of weakly left E-ample semigroups, as studied by Fountain, Gomes, Gould and Lawson, amongst others.  In the present article, we will survey the historical development of the study of left restriction semigroups, from the `weakly left E-ample' perspective, and sketch out the basic aspects of their theory.

From right PP monoids to restriction semigroups: a survey

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E-unitary covers for inverse semigroups

Cohomology of inverse semigroups

A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl*V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators *, **, ?, =, = n , and P do not preserve decidability. In addition, several joins, including A V U, are shown to be undecidable.

https://www.ams.org/proc/2003-131-09/S0002-9939-03-06860-6/S0002-9939-03-06860-6.pdf

On the extension problem for partial permutations

/pdf/groups-semilattices-and-inverse-semigroups-ii-4z7nc37eez.pdf

D. B. McAlister

Papers

Groups, semilattices and inverse semigroups. II