Imbedding Infinitely Distributive Lattices Completely Isomorphically Into Boolean Algebras

Abstract

It is well known that any distributive lattice can be imbedded in a Boolean algebra ([1], [2], [4] and others). This imbedding is in general only finitely isomorphic in the sense that the imbedding preserves finite sums (supremums) and finite products (infimums) (but not necessarily infinite ones). Indeed, in order to be able to be imbedded into a Boolean algebra completely isomorphically (i.e. preserving every supremum and infimum) a distributive lattice L must satisfy the infinite distributive law, as the infinite distributivity holds in Boolean algebras. The main purpose of this paper is to prove that the converse is also true, that is, any infinitely distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 6). Since we show, on the other hand, that any relatively complemented distribu tive lattice is infinitely distributive (Theorem 2), Theorem 6 implies that every relatively complemented distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 4).