Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α −1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.
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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.
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We give necessary and sufficient conditions for various vertex-transitivity of Cayley graphs of the class of completely 0-simple semigroups and its several subclasses. Moreover, the question when the Cayley graphs of completely 0-simple semigroups are undirected is considered.
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In this paper, completely regular endomorphisms of the join of split graphs are investigated. We give conditions under which all completely regular endomorphisms of the join of two split graphs form a monoid.
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In this paper we introduce perfectly supportable semigroups and prove that they are σ-discrete in each Hausdorff shiftinvariant topology. The class of perfectly supportable semigroups includes each semigroup S such that FSym(X) ⊂ S ⊂ FRel(X) where FRel(X) is the semigroup of finitely supported relations on an infinite set X and FSym(X) is the group of finitely supported permutations of X.
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