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Semi-automorphisms of transformation semigroups. (English) Zbl 0537.20037
A semi-automorphism of a semigroup S is any bijection of S such that \((aba)\phi =a\phi b\phi a\phi\) for all \(a,b\in S.\) For rings, additional conditions are placed on \(\phi\) (including that it be additive). These and related maps arise quite naturally in the course of various investigations and the author gives an interesting historical account of the development and application of these concepts in the introduction. He then goes on to determine the semi-automorphisms of certain types of transformation semigroups. A subsemigroup S of \({\mathcal P}_ X\), the semigroup of all partial transformations on a set X, is said to cover X if for each \(x\in X\), some constant idempotent of S has \(range\quad \{x\}\) and S extremally covers X if S contains each constant function whose domain is a singleton as well as each constant function whose domain is all of X. Finally, S is 2-transitive on X if for \(x,y,a,b\in X\) with \(x\neq y\), then \(x\alpha =a\) and \(y\alpha =b\) for some \(\alpha\in S\). The author then proves that if S is 2-transitive and extremally covers X, then every semi-automorphism of S is, in fact, an inner automorphism. Here, an automorphism \(\phi\) of S is inner if \(\alpha \phi =h^{- 1}\alpha h\) for all \(\alpha\in S\) where h is some bijection of X. In particular, h need not belong to S. He also proves that if S is a 2- transitive inverse subsemigroup of \({\mathcal I}_ X\), the full symmetric inverse semigroup on X, which covers X, then a semi-automorphism of S is either an inner automorphism or the composition of an inner automorphism with the map which sends each element to its inverse. In the last portion of the paper, the author conjectures that a half-automorphism of any 2- transitive transformation semigroup S, extremally covering X, must be an automorphism and he proves a lemma which, if the conjecture is to be settled affirmatively, is likely to be useful in settling it. A half- automorphism of S is any bijection \(\phi\) of S such that for all a,\(b\in S\), (ab)\(\phi\) is either \(a\phi\) \(b\phi\) or \(b\phi\) \(a\phi\).
Reviewer: K.D.Magill, jun

20M15 Mappings of semigroups
20M20 Semigroups of transformations, relations, partitions, etc.
01A60 History of mathematics in the 20th century
20-03 History of group theory
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