Automorphisms and range families of transformation semigroups
Inessa Levi · Bulletin of the Australian Mathematical Society · 1985
The problem of describing a l l automorphisms of a given semigroup of transformations of a set X has interested a number of mathematicians in the past fifty years.In 1937 Schreier [10] showed that every automorphism of the full transformation semigroup Ty is innev (that i s , acts as a conjugation by some bijection of X).In 1952 Mal'cev [7] generalized this result by showing that every ideal of T« has only inner automorphisms.More recently Symons [11] showed that a l l automorphisms of any Gy-notfnal semigroup (that i s , invariant under a conjugation by any bijection of X) over a finite set X are inner, while Schein [9] produced the same result for Gy~norinal semigroups of one-to-one transformations over an infinite set X. (See [2] for the special base of Baer-Levi semigroups.)Chapters 2 and 3 of this thesis constitute a contribution towards the solution of the problem of describing a l l automorphisms of a given semigroup of transformations of an infinite set X . in Chapter 2 (see also [4]) 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 Gy-normal semigroup of transformations of an infinite set X has only inner automorphisms.Our purpose in Chapter 3 (see also [3]) i s to offer a complete description of a l l automorphisms of an arbitrary Croisot-Teissier