The semilattice tensor product of distributive lattices

Abstract

We define the tensor product A ⊗ B A \otimes B for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see [4], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice A ⊗ B A \otimes B is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of A ⊗ B A \otimes B and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices.