Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

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

On regular semigroups

1This book was published 1986 by the Akademie-Verlag Berlin, Volume 32 in the series “Mathematical Research”; Lector was Dr. Reinhard Hoppner. It was printed in the German Democratic Republic. It has been transferred into LATEX by Ulrich Thiemann. In connection with this translation also the notation for the direction used for the composition of homomorphisms has been changed (the first morphism is written to the right of the second one, etc.). So far most of the diagrams are still missing. Also an index is still missing. The errors and misprints in the original version have not yet been corrected. The bibliography has been extended about 1990 and not yet been updated – moreover, it may still contain many titles, which treat “partial operations” (which was the keyword for the search in the Zentralblatt), but which are not really concerned with partial algebras in the sense of this book. In order to get a better idea of the material in the book, the article: Tools for a Theory of Partial Algebras, which has been published in: General Algebra and Applications (Eds.: K.Denecke and H.-J.Vogel), Research and Exposition in Mathematics, Vol. 20, Heldermann Verlag Berlin, 1993, pp. 12–32, has been added to a revised introduction.

A Model Theoretic Oriented Approach to Partial Algebras

E-unitary inverse monoids and the Cayley graph of a group presentation

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

Free inverse monoids and graph immersions

This paper is concerned with a study of inverse monoids presented by a set X subject to relations of the form e i = f i , i ∈ I, where e i and f i are Dyck words, i.e. idempotents of the free inverse monoid on X. Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on X. In the finitely presented case the word problem is solved by using Rabin's theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored

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Inverse monoids, trees, and context-free languages

http://www.math.unl.edu/~mbrittenham2/papers/pdf/subgroups.free.ig.semigroups.pdf

John Meakin

Papers

E-unitary inverse monoids and the Cayley graph of a group presentation

Free inverse monoids and graph immersions

Inverse monoids, trees, and context-free languages

Subgroups of free idempotent generated semigroups need not be free

PSPACE-complete problems for subgroups of free groups and inverse finite automata