I have a concrete construction of the coproduct of distributive lattices and wondered if it was already known. Let A and B be distributive lattices. It is reasonably clear that their coproduct can be constructed as a tensor product with respect to their join-semilattice structure, using the notion of join-bilinear map. Even if this is not already known, it is an obvious application of the methods commonly used in the infinitary case for frames and suplattices. My concrete construction amounts to a description of when \/_i (a_i tensor b_i) <= \/_j (a'_j tensor b'_j) (*) I have shown that a necessary and sufficient condition for (*) is as follows. Let the j's range from 1 to n (we can assume wlog that the finite indexing set is a finite cardinal) and let D_n be the free distributive lattice on n generators. Let ~: D_n -> D_n be the operation that interchanges meets and joins (i.e. the lattice homomorphism from D_n to (D_n)^op that is the identity on generators). Then (*) holds iff for each i there is some phi_i in D_n such that a_i <= phi_i(a') b_i <= ~phi_i(b') (The appearance of ~ reflects the kind of topological argument about product spaces where unions in one component are balanced by intersections in the other.) Hence the coproduct of A and B is the set of formal expressions \/_i (a_i tensor b_i) (essentially, lists of elements of AxB) modulo the preorder <= as just defined. Steve Vickers Department of Pure Maths Faculty of Maths and Computing The Open University ----------- Tel: 01908-653144 Fax: 01908-652140 Web: http://mcs.open.ac.uk/sjv22 15-Sep-2002 15:19:44 -0300,2063;000000000000-00000000