Progressive or linear or ... monoids?

1. Is the quasivariety of monoids generated by the groups and the free monoids finitely based? That is, is there a finite set of universal Horn formulas entailing the common universal Horn theory of groups and free monoids? In other words, what do groups and free monoids have in common, besides being monoids? Apart from the (equational) axioms for monoids, the only members of that theory I can think of are xy=x -> y=1 and yx=x -> y=1. 2. How different is the abelian case? More or fewer axioms? Vaughan Pratt