Rank Properties in Semigroups of Mappings

Abstract

The rank of a finite semigroup S is defined as r(S) = min{|A|: ‹A› = S}. If S is generated by its set E of idempotents or by its set N of nilpotents, then the idempotent rank ir(S) and the nilpotent rank nr(S) are given by ir(S) = min{|A|:A ⊆ E and ‹A› = S} and nr(S) = min{|A|:A ⊆ n and ‹A› = S} respectively; these are potentially different from r(S). If Singn is the semigroup of all singular self-maps of {1, …, n} then r(Singn) = ir(Singn) = 1/2n(n−1). If SPn is the inverse semigroup of all proper subpermutations of {1, …, n} then r(SPn) = nr(SPn) = n + 1.