Automorphisms and range families of transformation semigroups
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
The problem of describing all automorphisms of a given semigroup of transformations of a set X has interested a number of mathematicians in the past fifty years. J. Schreier showed that all automorphisms of the full transformation semigroup T“ are inner, and A. Mal'cev showed that the same property holds for any ideal of T“. More recently J. Symons showed that all automorphisms of any G“-normal semigroup over a finite set x are inner, while B.M. Schein produced the same result for G“-normal semigroups of 1-1 transformations over an infinite set X. Chapters 2 and 3 of this thesis constitute a contribution toward the solution of the problem of describing all automorphisms of a given semigroup of transformation of an infinite set X. In Chapter 2 we extend the well-known result from group theory, namely that any normal group of bijections of an infinite set X has only inner automorphisms, to an analogous one in semigroup theory. We show that any G“-normal semigroup of transformations of an infinite set X has only inner automorphisms. In Chapter 3 (which is a joint work with K.C. O'Meara and G.R. Wood) we give the description of all automorphisms of an arbitrary Croisot-Teissier semigroup. They offer a rich variety, from inner to "locally" inner, to thoroughly outer. We also present a description of Green's relations on Croisot-Teissier semigroups. In Chapter 4 we define a normal subset of the power set P“of an infinite set X. We characterize all normal subsets of P“ which serve as sets of ranges of semigroups of total transformations of X. In Chapter 5 for a particular normal subset of P“ we give necessary and sufficient conditions for an order-automorphism to be determined by a bijection of X (that is, induced). We then characterize those normal subsets of P“ for which all order-automorphisms are induced.