Graph expansions of unipotent monoids

Margolis and Meakin use the Cayley graph of a group presentation to constru ct E-unitary inverse monoids [11]. This is the technique we refer to as gra ph expansion. In this paper we consider graph expansions of unipotent monoi ds, where a monoid is unipotent if it contains a unique idempotent. The mon oids arising in this way are E-unitary and belong to the quasivariety of we akly left ample monoids. We give a number of examples of such monoids. We s how that the least unipotent congruence on a weakly left ample monoid is gi ven by the same formula as that for the least group congruence on an invers e monoid and we investigate the notion of proper for weakly left ample mono ids. Using graph expansions we construct a functor F-e from the category U of un ipotent monoids to the category PWLA of proper weakly left ample monoids. T he functor F-e is an expansion in the sense of Birget and Rhodes [2]. If we equip proper weakly left, ample monoids with an extra unary operation and denote the corresponding category by PWLA(0) then, regarded as a functor U --> PWLA(0), F-e is a left adjoint of the functor F-sigma :PWLA(0) --> U th at takes a proper weakly left ample monoid to its greatest unipotent image. Our main result uses the covering theorem of [8] to construct free weakly l eft ample monoids.