The semigroup of one-to-one transformations with finite defects
Inessa Levi, Boris M. Schein · Glasgow Mathematical Journal · 1989
Let be the semigroup of all total one-to-one transformations of an infinite setX.For an ƒ ∈ let thedefectofƒdefƒ, be the cardinality ofX – R(ƒ), whereR(ƒ) = ƒ(X) is the range of ƒ. Then is a disjoint union of the symmetric group xonX, the semigroupSof all transformations in with finite non-zero defects and the semigroup Ā of all transformations inSwith infinite defects, such that S U Ā and Ā are ideals of . The properties of xand 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 semigroupS.It is not difficult to see thatSis left cancellative (we compose functions ƒ,ginSas ƒg(x) = ƒ(g(x)), forx∈X) and idempotent-free. All automorphisms ofSare inner [4], that is of the form ƒ →hƒhfh-1ƒ ∈S,h∈ x.