A combinatorial property of finite full transformation semigroups

Abstract

Synopsis Let E be the set of idempotents in the semigroup Sing n of singular self-maps of N = {1, …, n }. Let α ∊ Sing n . Then α ∊ E 2 if and only if for every x in im α the set xα −1 either contains x or contains an element of (im α)′. Write rank α for |im α| and fix α for |{ x ∊ N : xa = x }|. Define ( x , xα , xα 2 ) to be an admissible α- triple if x ∊ (im α)′, x α 3 ≠ x α 2 . Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E 3 if and only if