Implication Algebras and the Metropolis - Rota Axioms for Cubic Lattices Original Research Article

This paper is motivated by a result of Metropolis and Rota on an algebraic characterization of the lattice of faces of the n-cube (cubic lattice). Although their proof relies on an inductive argument, the axioms are independent of the dimension n. The question of how to extend this theory to include infinite cubic lattices was left open. We develop an extended characterization theory of cubic lattices of arbitrary dimension by adding three axioms (completeness, atomicity, and coatomicity) to those of Metropolis and Rota. The proof of our main theorem depends on the introduction of the cubic implication algebra, which is shown to satisfy Abbott′s axioms for implication algebras. These algebras were first developed to characterize semi-Boolean algebras and Boolean algebras.