I haven't had a chance yet to follow up on Mike Barr's thoughts, but hopefully will when I get back from the Jonsson symposium. Meanwhile here's a message from Pepe Meseguer giving some key references, which would appear to be prerequisite reading for anyone formulating ideas in this area. -v All of this has been done for ordered algebras by Steve Bloom, who defined ordered theories and proved a Birkhoff variety theorem in "Varieties of ordered algebras" JCSS 13 (1976) 200-212, and for (chain) continuous ordered algebras by me, who used continuous theories, and proved a completeness theorem and two different Birkhoff variety theorems in "Varieties of Chain-Complete Algebras" J. Pure Appl. Algebra, 19 (1980) 347-383, and in "A Birkhoff-like theorem for algebraic clases of interpretations of program schemes" in "Formalization of Programming Concepts" J. Diaz and I. Ramos (eds.) Springer LNCS 107, (1981) 151-168.