### Infinite injective transformations whose centralizers have simple structure

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...