Tight Galois connections and complete distributivity

Abstract

and discusses its relations with the property of complete distributivity in complete lattices. A procedure for constructing Galois connections between complete lattices is presented. The Galois connections constructed by this procedure are called tight Galois connections, and are characterized as those which satisfy certain identities. All closure operations on complete lattices are obtainable from tight Galois connections. If either of the complete lattices involved in a Galois connection is completely distributive, then the Galois connection is tight. Consequently, all Galois connections constructed by the known procedure of Birkhoff are tight. The identity mapping from a complete lattice to its dual lattice always determines a Galois connection; this Galois connection is tight if and only if the complete lattice is completely distributive. This last observation leads to a characterization of completely distributive complete lattices solely in terms of the partial ordering on them. It also provides new insight into the structure of these lattices, and enables us to prove a representation theorem which is considerably more economical than the one previously known. 2. Definitions and notations. If F is a family of subsets of a set, the intersection of F is denoted by HF, and the union of F is denoted by EF. If L is a complete lattice, then every subset K of L has a meet, which is denoted by nK, and a join, which is denoted by UK.