Complete semigroups (Q1964726)

Let \(S\) be a finite monoid. For \(a\in S\), let \(L^+(a)=\{x\in S\mid xa=x\}\) and \(\delta(a)=|L^+(a)|/|Sa|\). Since \(L^+(a)\subseteq Sa\), \(\delta(a)\) is a rational number such that \(0\leq\delta(a)\leq 1\), with \(\delta(a)=1\) if and only if \(a\) is an idempotent. The finite monoid \(S\) is said to be complete if for every non-idempotent \(a\) of \(S\), \(a\) is in the subsemigroup of \(S\) generated by \(\{b\in S\mid\delta(b)>\delta(a)\}\). Thus, \(S\) is complete if and only if \(S\) is generated by its idempotents. Let \(\text{Sing}_n\) and \(\text{SP}_n\) be the semigroups of singular transformations and of singular partial transformations, respectively, of an \(n\)-element set. Using the concepts introduced before, it is shown via an inductive approach that the monoids \(\text{Sing}^1_n\) and \(\text{SP}^1_n\) are complete. Also the monoid of endomorphisms of a finite chain is shown to be complete.