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

Published online by Cambridge University Press:  18 May 2009

Inessa Levi  and
Boris M. Schein
Inessa Levi
Affiliation:
Department of Mathematics, University of Louisville, Lousville, Kentucky 40292
Boris M. Schein
Affiliation:
Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
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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 x ∈ X) and idempotent-free. All automorphisms of S are inner [4], that is of the form ƒ → hƒhfh-1 ƒ ∈ S, h ∈ x.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 2 , May 1989 , pp. 243 - 249
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

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