Transformation semigroups: past, present and future
type: Item
instance of: scholarly article
zbMATH Open document ID: 0982.20055
publication date: 2002-04-02T00:00:00Z
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20-02
zbMATH DE Number: 1584847
zbMATH Keywords: semigroups of partial transformations
zbMATH Keywords: transformation semigroups
zbMATH Keywords: compatible subsemigroups
zbMATH Keywords: Vagner-Preston theorem
zbMATH Keywords: inverse semigroups
zbMATH Keywords: symmetric inverse semigroups
zbMATH Keywords: ideals
zbMATH Keywords: Baer-Levi semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: nilpotents
zbMATH Keywords: congruences
author: R. P. Sullivan
MaRDI profile type: Publication
title: Transformation semigroups: past, present and future
review text: This is an expository paper on the theory of transformation semigroups written by someone who has contributed a great deal to that theory. A transformation semigroup is any subsemigroup of the semigroup \(P(X)\) of all partial transformations, under composition, of a set \(X\).NEWLINENEWLINENEWLINEThe introduction and notation and terminology comprise the first two sections. Section 3 is devoted to a discussion of what has been done up to the present time in transformation semigroups and is divided into eight subsections. These are respectively entitled Embeddings, Ideals, Green's Relations, Idempotents, Combinatorics, Nilpotents, Morphisms, and Congruences.NEWLINENEWLINENEWLINEFor an example of the type of result discussed in the Embedding subsection we turn to a result of Boris Schein. A subsemigroup \(Q\) of \(P(X)\) is compatible if \(\alpha\cup\beta\in P(X)\) for all \(\alpha,\beta\in Q\). Schein's result is that a semigroup \(S\) is isomorphic to a compatible subsemigroup of \(P(X)\) if and only if \(xy^2=xy\) and \(xyz=xzy\) for all \(x,y,z\in S\). The well known Vagner-Preston theorem on inverse semigroups is also discussed. That is, any inverse semigroup can be embedded in \(I(X)\), the symmetric inverse semigroup on an appropriate set \(X\).NEWLINENEWLINENEWLINEAs for the subsection on Ideals, A. I. Malcev's results on the ideals of \(T(X)\) the full transformation semigroup on a set \(X\) are discussed. The author goes on to discuss results obtained by various authors, including his own, on various types of transformation semigroups, including the Baer-Levi semigroups. In the subsection on Green's Relations, the author presents the well known results concering \(T(X)\) as well as results he and M. Paula O. Marquez-Smith have recently obtained for \(P(X)\).NEWLINENEWLINENEWLINEIt was John Howie who first looked carefully at \(E(X)\), the subsemigroup generated by the idempotents (other than the identity) of \(T(X)\). His work is discussed and it is pointed…
reviewed by: jun. Kenneth D. Magill
description: scientific article