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Infinite injective transformations whose centralizers have simple structure

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Konieczny J.

Open Mathematics

2011

tom 9

nr 1

23-35

For an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X under function composition. For α ∈ Γ(X), let C(α) = {β ∈ g/g(X): αβ = βα} be the centralizer of α in Γ(X). The aim of this paper is to determine those elements of Γ(X) whose centralizers have simple structure. We find α ∈ (X) such that various Green's relations in C(α) coincide, characterize α ∈ Γ(X) such that the $$ \mathcal{J} $$-classes of C(α) form a chain, and describe Green's relations in C(α) for α with so-called finite ray-cycle decomposition. If α is a permutation, we also find the structure of C(α) in terms of direct and wreath products of familiar semigroups.

Ograniczanie wyników

1 Open Mathematics

1 Konieczny J.

1 2011

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Infinite injective transformations whose centralizers have simple structure