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The semigroup of one-to-one transformations with finite defects

  • Inessa Levi (a1) and Boris M. Schein (a2)

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Let be the semigroup of all total one-to-one transformations of an infinite set X. For an ƒ ∈ let the defect of ƒ def ƒ, be the cardinality of X – R(ƒ), where R(ƒ) = ƒ(X) is the range of ƒ. Then is a disjoint union of the symmetric group x on X, the semigroup S of all transformations in with finite non-zero defects and the semigroup Ā of all transformations in S with infinite defects, such that S U Ā and Ā are ideals of . The properties of x and Ā have been investigated by a number of authors (for the latter it was done via Baer-Levi semigroups, see [2], [3], [5], [6], [7], [8], [9], [10] and note that Ā decomposes into a union of Baer–Levi semigroups). Our aim here is to study the semigroup S. It is not difficult to see that S is left cancellative (we compose functions ƒ, g in S as ƒg(x) = ƒ(g(x)), for xX) and idempotent-free. All automorphisms of S are inner [4], that is of the form ƒ → hƒhfh-1 ƒ ∈ S, hx.

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References

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1.Clifford, A. H. and Preston, G. B., Algebraic theory of semig roups, Math Surveys No. 7, Amer. Math. Soc, Providence, R.I., Vol. I (1961).
2.Levi, I., Schein, B. M., Sullivan, R. P. and Wood, G. R., Automorphisms of Baer-Levi semigroups, J. London Math. Soc. (2) 28 (1983), 492495.
3.Levi, I. and Wood, G. R., On maximal subsemigroups of Baer-Levi semigroups, Semigroup Forum, 30 (1984), 99102.
4.Levi, I., Automorphisms of normal transformation semigroups, Proc. Edinburgh Math. Soc., 28 (1985), 185205.
5.Lindsey, D. and Madison, B., The lattice of congruences on Baer-Levi semigroup, Semigroup Forum 12 (1976), 6370.
6.Mielke, B. W., Regular congruences on Croisot-Teissier and Baer-Levi semigroups, J. Math. Soc. Japan 24 (1972), 539551.
7.Mielke, B. W., Regular congruences on a simple semigroup with a minimal right ideal, Publ. Math. Debreceen 20 (1973), 7984.
8.Mielke, B. W., Completely simple congruences on Croisot-Teissier semigroups, Semigroup Forum 9 (1975), 283293.
9.Schein, B. M., Symmetric semigroups of one-to-one transformations, Second all-union symposium on the theory of semigroups, summaries of talks, Sverdlovsk (1979), 99 (Russian).
10.Shutov, E. G., Semigroups of one-to-one transformations, Dokl. Akad. Nauk. SSSR 140 (1961), 10261028.

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