THE STRUCTURE OF END(Tn)

The full transformation semigroups Tₙ, where n ∈ ℕ, consisting of all maps from a set of cardinality n to itself, are arguably the most important family of finite semigroups. This article investigates the endomorphism monoid End(Tₙ) of Tₙ. The determination of the elements of End(Tₙ) is due Schein and Teclezghi. Surprisingly, the algebraic structure of End(Tₙ) has not been further explored. We describe Green’s relations and extended Green’s relations on End(Tₙ), and the generalised regularity properties of these monoids. In particular, we prove that H = L ⊆ R = D = J (with equality if and only if n = 1); the idempotents of End(Tₙ) form a band (which is equal to End(Tₙ) if and only if n = 1) and also the regular elements of End(Tₙ) form a subsemigroup (which is equal to End(Tₙ) if and only if n ≤ 2). Further, the regular elements of End(Tₙ) are precisely the idempotents together with all endomorphisms of rank greater than 3. We also provide a presentation for End(Tₙ) with respect to a minimal generating set.