On the growth of generating sets for direct powers of semigroups

For a semigroup S its d -sequence is d ( S )=( d 1 , d 2 , d 3 ,…), where d i is the smallest number of elements needed to generate the i th direct power of S . In this paper we present a number of facts concerning the type of growth d ( S ) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.