Dear Vaughan A related question, originating with the problem of the quality of the truth value space in M- sets, led me to the discovery of an equational class that some years later, Peter Freyd rediscovered from an entirely different point of view. What is the Structure of the union of the "variety" groups with the variety of commutative monoids ? The motivation was the common feature of the Heyting algebra of right ideals of a monoid in either class, and the partial answer is in my "Taking categories seriously", reprinted in TAC. Not only do we have to think about the meaning of "union" and about the analogy with closed subschemes (probably the source of the term "variety") but also about the fact that the inclusion of groups in monoids is more "open" than closed and is certainly not a (sub) variety even though both categories are algebraic. Again analogously, the Structure 2-functor, adjoint to Semantics does not carry full inclusions to surjective interpretations, Thus in particular in my example like others, the algebraic theory that results has more operations, not only more equations. It may be true in your case as well. Bill Quoting Vaughan Pratt <pratt@cs.stanford.edu>
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