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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

##### MSC:
 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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##### References:
  G. Ancochea: On semi-automorphisms of division algebras. Ann. of Math., 48 (1947) 147-153. · Zbl 0029.10703  L. Childs: A Concrete Introduction to Higher Algebra. Springer-Verlag, 1979. · Zbl 0397.00002  A. H. Clifford, G. B. Preston: The Algebraic Theory of Semigroups. Math. Surveys, No. 7, Amer. Math. Soc., Providence, RI, Volume I (1961), Volume 2 (1967). · Zbl 0111.03403  J. Dieudonné: On the automorphisms of the classical groups. Mem. Amer. Math. Soc., No. 2, 1951. · Zbl 0042.25603  F. Dinkines: Semi-automorphisms of symmetric and alternating groups. Proc. Amer. Math. Soc., 2 (1951) 478-486. · Zbl 0042.25404  A. E. Evseev, N. E. Podran: Semigroups of transformations generated by idempotents with given projection characteristics. Izv. Vyss. Ucebn. Zaved. Mat., 12 (103) 1970, 30-36. · Zbl 0668.20058  I. N. Herstein, M. F. Ruchte: Semi-automorphisms of groups. Proc. Amer. Math. Soc., 9 (1958) 145-150. · Zbl 0080.01703  J. M. Howie: The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc., 41 (1966) 707-716. · Zbl 0146.02903  Loo-Keng Hua: On the automorphisms of a sfield. Proc Nat. Acad. Sci. U.S.A., 35 (1949) 386-389. · Zbl 0033.10402  N. Jacobson: Isomorphisms of Jordan rings. Amer. J. Math., 70 (1948) 317-326. · Zbl 0039.02801  N. Jacobson, C. E. Rickart: Jordan homomorphisms of rings. Trans. Amer. Math. Soc., 69 (1950) 479-502. · Zbl 0039.26402  N. Jacobson: Basic Algebra. I, W. H. Freeman and Co., 1974. · Zbl 0284.16001  I. Kaplansky: Semi-automorphisms of rings. Duke Math. J., 14 (1947) 521-525. · Zbl 0029.24801  K. D. Magill: Semigroup structures for families of functions. I, J. Austral. Math. Soc., 7 (1967) 81-94. · Zbl 0163.17104  W. R. Scott: Half-homomorphisms of groups. Proc Amer. Math. Soc., 8 (1957) 1141- 1144. · Zbl 0080.24504  W. R. Scott: Group Theory. Prentice-Hall, 1964. · Zbl 0126.04504  W. R. Scott: Semi-isomorphisms of certain infinite permutation groups. Proc. Amer. Math. Soc., 21 (1969) 711-713. · Zbl 0175.29902  L. N. Sevrin: Semi-isomorphisms and lattice isomorphisms of semigroups with a cancellation law. Dokl. Akad. Nauk SSSR, 171 (1966) 296-298; translated as Sov. Math. Dokl., 7 (1966) 1491-1493. · Zbl 0189.02201  L. N. Sevrin: Semi-isomorphisms of semigroups with cancellation law. Izv. Akad. Nauk SSSR, Ser. Mat. 31 (1967) 957-964.  R. P. Sullivan: Automorphisms of transformation semigroups. J. Austral. Math. Soc., 20 (Series A) 1975, 77-84. · Zbl 0318.20042  R. P. Sullivan: Automorphisms of injective transformation semigroups. · Zbl 0507.20037  J. S. V. Symons: Automorphisms of transformation semigroups. Ph. D. Thesis, University of Western Australia, 1973. · Zbl 0326.20058  M. Weinstein: Examples of Groups. Polygonal Pub. House, 1977. · Zbl 0359.20001
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