Subordinations, closed relations and compact Hausdorff spaces

We introduce the concept of a subordination, which is dual to the well-known concept of a precontact on a Boolean algebra. We develop a full categorical duality between Boolean algebras with a subordination and Stone spaces with a closed relation, thus generalizing the results of [14]. We introduce the concept of an irreducible equivalence relation, and that of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. We prove that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras, thus providing a “modal-like” alternative to de Vries duality.