Dear Bob: One thing I didn't say anything about in my message last night is what I meant by HSP. The P is standard. As for S it is pretty clear from what I said that S stands for strict (=extremal) subobjects. It follows that H stands for surjective images. This probably contradicts expectation, so a few words of explanation are necessary. First place, H and S must determine each other as a factorization system. Also H has to be closed under products. I'll come back to that. Let us look at S. If S just stands for subobjects, then even such a simple relation as x < x+y isn't inherited by subobjects and so doesn't define an HSP class. This is clearly not what is wanted. Thus only strict subojects are wanted. This forces H to include all surjective images and then it is obvious that H/S give a factorization system. Also H is closed under products. (In fact, it is closed under pullbacks, which implies there is a regular epi/mono factorization system, but not every morphism in H is regular.) H is also invariant under the endofunctor * on the category of posets that takes each poset to its opposite. Now let Th be one of Vaughn's theories. Then there are operations of arity (m,n) for which a model is a poset X equipped with a monotone map X*^m x X^n ---> X. Then these will be subject ot some constraints (equations and inequalities) and we get a category of models. Suppose that X ---> Y is a morphism of models and we factor that in the category of posets as X --->> Z >---> Y where the first map is surjective and the second is strong monic in Pos. We have the following rectangle: X*^m x X^n --->> Z*^m x Z^n ---> Y*^m x Y^n ! ! ! ! ! ! ! ! ! ! ! ! ! v v v X -------------> Z >-----------> Y where the outer vertical arrows are the given sturcture maps and the middle one exists by virtue of the factorization system. This shows that in the category of algebras there is a factorization into surjections/S-monomorphisms, the latter being strict in the underlying category Pos. They are not necessarily strict in the category of models. For example in the category of po-monoids, the inclusion of N into Z (both with usual ordering) is strict in Pos but not in the category since it is epic and a strict-monic/epic is an isomorphism. The fact that Pos is not a regular category makes me think it unlikely that anything could be done with a regular-epic/monic factorization, even if that were desirable. I guess that regular epics are closed under finite (but probably not infinite) products, so the first part of the analysis could be carried out for finitary theories. But even if it were possible, it would mean that only equations, not inequalities could be specified and that is certainly not what is wanted. I haven't checked out any of the details in this case. Regards, Mike