Rank and Status in Semigroup Theory

Abstract

For each generating set A of a finite semigroup S the integer ∆(A) is defined as the least n for which every element of S is expressible as a product of at most n elements of A. The status of S is defined as the least value of |A|∆(A) among generating sets of A. Some general bounds are obtained, and the notion is explored in more detail for certain well understood classes of semigroups.