Inverse monoids, regarded as algebras with a binary operation (the semigroup multiplication), a unary operation (a ? a? ), and a nullary operation with image 1, form a variety. Consequently, for any nonempty set X, the free inverse monoid F IM(X) exists. Over 30 years ago, Scheiblich and Munn provided a beautiful description of F IM(X); it is proper, and as such can be represented as a McAlister P semigroup P(G, X , Y). In fact, G is the free group F G(X) on X and Y is a semilattice of certain finite subsets of F G(X). We recall that a monoid is left ample (formerly, left type A) if it is a submonoid of some symmetric inverse monoid IX closed under the unary operation ? ? ?+ = Idom ? . Right ample monoids are defined dually; a monoid is ample if it is both left and right ample. (Left) ample monoids have abstract characterisations using the generalisations R? and L? of Green's relations R and L; they form quasivarieties of algebras. As such, the free left ample monoid F LAM(X) and free ample monoid F AM(X) exist. In the 80's Fountain characterised F LAM(X); it is a submonoid of F IM(X). Surprisingly, ample monoids can be more awkward than left ample monoids. For example, they have no nice representation theory as subalgebras of IX corresponding to that in the onesided case. Recently, the speaker, together with Fountain and Gomes, has shown that F AM(X) is also a submonoid of F IM(X).
