Nilpotents and congruences on semigroups of transformations with fixed rank

Abstract

In1998, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if ∆m denotes the Malcev congruence on Pm, then Pm/∆m is congruence-free for any infinite m. In this paper, we describe the products of nilpotentes in Pm, when m is non-regular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.