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 non-empty 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 sub-monoid 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 quasi-varieties 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).