Duality for Boolean Equalities

0. Notation. The basic notation for Boolean algebras is the standard one (for instance, see [l]). Throughout this paper, the letter O will denote the two-element Boolean algebra; whenever a topology is assumed to be defined on 0, then this topology is the discrete one. If 73 is a Boolean algebra, then a valuation of B is a homomorphism from 73 onto 0. We make precise the terminology we shall be using in connection with the Stone duality theory for Boolean algebras. A Boolean space is a compact Hausdorff space whose topology is generated by the clopen sets. If X is a Boolean space, then the dual algebra of X is the Boolean algebra of clopen sets of X. If 73 is a Boolean algebra, then the dual space of 73 is the Boolean space X of all valuations of 73 (the topology on X is the topology generated by all sets of the form {v: vp = 1} where p belongs to B).