The Semilattice Tensor Product of Distributive Lattices

We define the tensor product A ® S for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see 14], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice A <3 B is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of A C defined by hb(a) = f(a, b) are homomorphisms for all aEA and b SB. Definition 2.2. Let A and B be semilattices. A semilattice C is a tensor product of A and B if there is a bihomomorphism f:A x B —► C such that C is generated by f(A x B) and for any semilattice D and any bihomomorphism g:A x B —► D there is a homomorphism h:C—>D satisfyingg = hf. Note that since f(A x B) generates C, the homomorphism h is necessarily unique. Theorem 2.3. Let A and B be semilattices. Then a tensor product of A and B exists and is unique up to isomorphism. Proof. Let K be the free semilattice on A x B and let w be the canonical inclusion map of A x B into K. Let p be the set of all ordered pairs of the Received by the editors November 4, 1974. AMS (MOS) subject classifications (1970). Primary 06A20, 06A35. „ Copyright © 1976. American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use