On the large rank of certain semigroups of transformations on a finite chain

Abstract

The large rank of a finite semigroup [Formula: see text] is the least number [Formula: see text] such that every subset of [Formula: see text] with [Formula: see text] elements generates [Formula: see text]. This paper obtains the large ranks of [Formula: see text] and [Formula: see text], the semigroups of singular transformations, injective partial and partial transformations on a finite chain [Formula: see text], which preserve or reverse the order, respectively. As a consequence, we obtain the large ranks of [Formula: see text] and [Formula: see text], the semigroups of injective order-preserving partial transformations and order-preserving partial transformations on [Formula: see text], respectively.