Idempotents, regular elements and sequences from finite semigroups (Q1356419)

The paper proves a combinatorial result, the likes of which have proved useful in finite semigroup theory, that if \(n\) is the number of non-idempotent elements of a finite semigroup \(S\) then any product of length \(2^n\) has an idempotent factor (regarding the product as a word). The proof is a straightforward induction argument and examples which exploit properties of Zimin words show the bound of \(2^n\) is best possible. From this they calculate explicitly the integer \(b(n)\) such that for every semigroup of \(n\) elements any product of length \(b(n)\) has an idempotent factor. The value of \(b(n)\) is of the order of \(4^{n/3}\), the precise value depends on \(n\pmod 3\). The proofs allow similar results to be derived for other semigroup classes and element types.