Lattices, Closure Operators, and Galois Connections

Abstract

Lattices and semilattices are developed, both as partially ordered sets where every pair of elements has a least upper bound and/or a greatest lower bound, and as algebraic structures, and various completeness conditions they can satisfy are examined.Such structures often arise from closure operators on sets, and this concept is developed.An insufficiently well known source of closure operators, which we develop, is the concept of a Galois connection between two sets. In the case for which the concept was named, the two sets are, respectively, the elements of a finite separable normal field extension, and the automorphisms of that extension. Another of the many examples noted relates the set of models of a formal language, and the set of propositions in that language.