RECTANGULAR PRODUCTS OF LATTICES

Given lattices L1 and L2, their rectangular product L1□L2 is the set of ordered pairs (a,b) ε{lunate} L1 x L2, with a and b not zero, together with (0,0). Alternatively: L1□L2 = 1 ⊕ ((L1β{0}) × (L2β{0})). If L1 and L2 are face lattices of convex polytopes, then L1□L2 is the lattice of faces of the (topological) product of these polytopes. If L1 and L2 are concept in the sense of Wille, then L1□L2 is the concept lattice of the semiproduct of the underlying contexts. In this article, properties which are preserved (or not preserved) by rectangular products are discussed, and necessary and sufficient conditions are given for a lattice to be a rectangular product. © 1990.