Studia Scientiarium Mathematicarum Hungarica 15. (1980)

1-3. szám - Sullivan R. P.: Automorphisms of injective transformation semigroups

Studia Scientiarum Mathematicarum Hungarica 15 (1980), 1—4 . AUTOMORPHISMS OF INJECTIVE TRANSFORMATION SEMIGROUPS by R. P. SULLIVAN Dedicated to the memory of James and Helen Walker 1. Introduction Suppose X is an arbitrary set. The normal subgroups of 18 x are listed in Theorems 10.8.8 and 11.3.4 [6] and their automorphism groups are described in Theorem 11.4.8 [6]. When X is finite, all <8X -normal transformation semigroups which are not groups are listed in [8] and [9]. Various examples of infinite ^-normal transformation semigroups are discussed in [7] but we do not know of any general result for semi­groups that would correspond to Baer’s Theorem [1]. In [7] we showed that if S is any -normal transformation semigroup contain­ing a constant, then every automorphism of S is inner and Aut (S) is isomorphic to 18 x. Here we first extend this result by showing that if S is any -normal trans­formation semigroup that contains an element with non-zero rank less than \X\ then S contains a constant. This generalizes a remark of Fitzpatrick and Symons [4] and reduces the problem of determining all those transformation semigroups which are defined on an infinite set X and which possess only inner automorphisms to those in which every non-zero element has rank equal to \X\. The most general result in this regard is that of [4] where the authors use Schreier’s Theorem [5] determining the automorphisms of 18 x to show that if S is any total transformation semigroup con­taining '8x then every automorphism of S is inner (see [2] for a special case). We shall extend this result by showing that any infinite 18 x -normal subsemigroup of Jx (called an injective transformation semigroup) containing Alt (X) possesses only inner automorphisms and deduce Schreier’s result as a consequence. This paper was completed while visiting the Mathematics Institute of the Hun­garian Academy of Sciences. I would like to thank Professor O. Steinfeld and his colleagues for their generous hospitality during my stay in Budapest. I also gratefully acknowledge the helpful comments of Professor B. Schein on an early draft of this paper. 2. The main results Notation will be that of [3] and [7]. In particular we recall that a transformation semigroup is any subsemigroup of Фx. The following generalizes a remark of Fitz­patrick and Symons [4]. 1980 Mathematics Subject Classification. Primary 20M20; Secondary 20M15. Key words and phrases. Automorphism, injective, transformation, semigroup. Studia Scientiarum Mathematicarum Hungarica 15 (1980)

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