A characterization of finite ternary algebras

A ternary algebra is a De Morgan algebra (that is, a distributive lattice with 0 and 1 and a complement operation that satisfies De Morgan's laws) with an additional constant Phi satisfying phi = <(phi)over bar>, (a + (a) over bar) + phi = a + (a) over bar, and (a * (a) over bar) * phi = a * (a) over bar. We provide a characterization of finite ternary algebras in terms of "subset-pair algebras," whose elements are pairs (X,Y) of subsets of a given base set epsilon, which have the property X boolean OR Y = epsilon, and whose operations are based on common set operations.