We sharpen Hájek’s Completeness Theorem for theories extending predicate product logic, Π∀. By relating provability in this system to embedding properties of ordered abelian groups we construct a universal BL-chain L in the sense that a sentence is provable from Π∀ if and only if it is an L-tautology. As well we characterize the class of lexicographic sums that have this universality property.

In this paper we examine the problem of decomposing the lexicographic product of a cycle with an empty graph into cycles of uniform length. We determine necessary and sufficient conditions for a solution to this problem when the cycles are of odd length. We apply this result to find necessary and sufficient conditions to decompose a complete equipartite graph into cycles of uniform length, in t...

We use the theory of lexicographic shellability to provide various examples in which the rank of the homology of a Rees product of two partially ordered sets enumerates some set of combinatorial objects, perhaps according to some natural statistic on the set. Many of these examples generalize a result of J. Jonsson, which says that the rank of the unique nontrivial homology group of the Rees pr...