Sequences with idempotent products from finite regular semigroups

We find the smallest integer (n) such that for every regular semigroup S of order n, every sequence of length (n) of elements of S contains a consecutive subsequence whose product is an -element, where = ‘idempotent’, ‘core’ and ‘subgroup and core’. For arbitrary semigroups of order n, we also find (n) where = ‘regular’, ‘group’, ‘core’, ‘regular and core’ and ‘subgroup and core’.