Semigroups of high rank

Abstract

By the rank r(S) of a finite semigroup S we shall mean the minimum cardinality of a set of generators ofS. For a group G, as remarked in [3], one has r(G)≦log2|G|, the bound being attained when G is an elementary abelian 2-group. By contrast, we shall see that there exist finite semigroups S for which r(S)≧|S| – 1. In the hope that it will not be considered too whimsical, we shall refer to a finite semigroup S of maximal rank (i.e. for which r(S) = |S|) as royal; a semigroup of next-to-maximal rank (i.e. for which r(S) = |S|–1) will be called noble.