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The Semigroup of Surjective Transformations on an Infinite Set

For an infinite set X, denote by Ω(X) the semigroup of all surjective mappings from X to X. We determine Green’s relations in Ω(X), show that the kernel (unique minimum ideal) of Ω(X) exists and determine its elements and cardinality. For a countably infinite set X, we describe the elements of Ω(X) for which the 𝒟-class and 𝒥-class coincide. We compare the results for Ω(X) with the corresponding results for other transformation semigroups on X.