The Horn formulae quoted by Vaughan can of course be generalized to the cancellation laws: xy = xz => y = z (and similarly on the other side). In fact the free monoids aren't needed: every free monoid embeds as a submonoid of a (free) group, and so satisfies all Horn formulae true in groups. Thus Vaughan is asking for the universal Horn theory of those monoids which are embeddable in groups. I'm fairly sure that the answer to this is known, but I can't find a reference for it. In the commutative case, it's easy to see that the cancellation law is all you need (the proof is essentially the same as the proof that every integral domain embeds in a field), but I vaguely remember that in the non-commutative case you need something more than the cancellation laws. Generalizing in an obvious way: can one characterize those categories which admit faithful functors to groupoids by a finite collection of Horn formulae? Peter Johnstone ---------------- On Fri, 24 Mar 2006, Vaughan Pratt wrote:
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