Transformation semigroups: past, present and future
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 out that analogous results for \(P(X)\) were obtained by Evseev and Podran and, independently, also by the author. B. Harris and L. Schoenfeld obtained results on the number of idempotents in \(T(X)\) for finite \(X\) and some of their results are also discussed. Howie's results on \(E(X)\) suggested a number of combinatorial problems and these have been the object of a considerable amount of investigation by Howie and his coauthors, and other authors as well. Many of these results are discussed in the subsection on Combinatorics.NEWLINENEWLINENEWLINEThe study of \(N(X)\), the subsemigroup of \(P(X)\) which is generated by the nilpotents of \(P(X)\) was initiated by the author. A number of results are discussed in the subsection on Nilpotents, many of which are concerned with necessary and sufficient conditions in order for an element of \(P(X)\) to belong to \(N(X)\). An automorphism \(\varphi\) of a subsemigroup \(S\) of \(P(X)\) is inner if \(\alpha\varphi=g\alpha g^{-1}\) for all \(\alpha\in S\) and some \(g\in G(X)\), the group of all bijections on \(X\). A subsemigroup \(S\) of \(P(X)\) is said to be \(G(X)\)-normal if \(g\alpha g^{-1}\in S\) for all \(\alpha\in S\) and all \(g\in G(X)\). The subsection on Morphisms contains a discussion of a number of results on the morphisms of \(G(X)\)-normal semigroups (and other transformation semigroups as well) many of which obtained by the author and his students and by Inessa Levi.NEWLINENEWLINENEWLINEThe final subsection of Section 3 is concerned with congruences. It is well known that A. I. Malcev described the complete lattice of congruences and this was no small task. Malcev's work is discussed as well as related work by other authors.NEWLINENEWLINENEWLINESection 4 contains seven subsections and these are all concerned with directions that further research in transformation semigroups might take. Finally, there are 154 references listed. Anyone interested in transformation semigroups should read this paper.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00028].
scientific article