Dear Colleagues, Recently, Bill Lawvere proposed parallel pairs of morphisms in an algebraic theory as the categorical concept of "equational law". We have just observed that this is, for many sorted theories, the first definition that makes sense! For example, the following variant of Birkhoff's Variety Theorem holds for many-sorted algebras: a full subcategory is equationally presentable (in Bill's sense) iff it is closed under products, subobjects, regular quotients and directed unions. If equation is "traditionally" understood as a pair of terms (elements of a free algebra of the given signature) of the same sort, then Birkhoff's theorem is not true: Example. Consider algebras on two sorts (say, a,b) with no operations - that is, consider the category Set x Set. Take the full subcategory V of all objects (A,B) such that either A is empty or B has at most 1 element. This is an HSP subcategory of Set x Set, but there does not exist any equation that only works with one of the sorts such that all the objects of V satisfy it. But in the theory which is a free completion of the discrete category {a,b} under finite products the parallel pair of projections p_2, p_3: a x b x b -> b specifies V. If, on the other hand, equations are understood as pairs of terms together with a many-sorted set of variables (encoding the universal quantification), the following example demonstrates that we are beyond the realm of finitary logic: Example. Consider algebras on infinitely many sorts with no operations. The full subcategory of all objects which are either subobjects of the terminal object, or have all but finitely many sorts empty, is an HSP class. But it is not closed under directed unions, and thus cannot be described in finitary logic. Best regards Jiri Adamek and Enrico Vitale