Idempotents, regular elements and sequences from finite semigroups Original Research Article

If n is the number of nonidempotent elements of a finite semigroup S, it is shown that each sequence of length 2n of elements of S contains a consecutive subsequence whose product is an idempotent element, and that 2n is the best possible among all finite semigroups with n nonidempotent elements. The proof remains valid if ‘idempotent’ is replaced by each of the words or phrases ‘regular’, ‘group’, ‘core’, ‘regular and core’ and ‘group and core’. The best bound, among all semigroups S with |S| = n, is also found, for semigroups and for monoids with or without a zero.