The structure of End(T_n)

2023/07/21 by Victoria Gould, Gould, Victoria, Ambroise Grau +3
Computer Science · Medicine · #20M05 #20M10 #20M20 #Cell Adhesion Molecules Research #FOS: Mathematics #Rings and Algebras (math.RA) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2307.11596

openalex publication_date 2023/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The full transformation semigroups T_n, 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(T_n) of T_n. The determination of the elements of End(T_n) is due Schein and Teclezghi. Surprisingly, the algebraic structure of End(T_n) has not been further explored. We describe Green's relations and extended Green's relations on End(T_n), 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(T_n) form a band (which is equal to End(T_n) if and only if n=1) and also the regular elements of End(T_n) form a subsemigroup (which is equal to End(T_n) if and only if n≤ 2). Further, the regular elements of End(T_n) are precisely the idempotents together with all endomorphisms of rank greater than 3. We also provide a presentation for End(T_n) with respect to a minimal generating set.

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