The variety generated by finite nilpotent monoids

Abstract

~. A finite semigroup S is said to be nil~otent if S has a zero and if S n = 0 for some positive integer n. The family of all finite nilpotent semigroups forms a variety of finite semigroups in the sense of the word used by Eilenberg (see [EIL], particularly Section VIII.2) that is, it is closed under finite direct products and division. Let us say that a finite monoid M is nilpotent if M [i] is a nilpotent semigroup. The family of all finite nilpotent monoids does not form a variety of finite monoids, since the direct product of two nilpotent monoids is not, in general, a nilpotent monoid. One can, however, consider the smallest variety V of finite monoids which contains all the M