Closure operators on sets and algebraic lattices

Closure operators are abundant in mathematics; here are a few examples. Given an algebraic structure, such as group, ring, field, lattice, vector space, etc., taking the substructure generated by a set, i.e., the least substructure which includes that set, is a closure operator. Given a binary relation, taking the relation with certain properties, such as reflexive, transitive, equivalence, etc., generated by the given relation, i.e., the least relation with the desired properties which includes the original relation, is a closure operator. The consequence operators in mathematical logic are closure operators. Topological spaces can be defined by Kuratowski closure operators. In this survey paper we have selected those facts about closure operators that we consider to be the most important ones. We present them in the “Ockham razor” spirit, i.e., under no more hypotheses than necessary. We have thus detected three levels of generality: A) closure operators on posets, B) closure operators on complete lattices, and C) closure operators on fields of sets P(A), which we tackle in this order. Level A, due to Bourbaki, is less known. However the fundamental property used in the applications of closure operators to various fields of mathematics is valid for arbitrary posets: the existence of the closure x̄ of each element x of the underlying set, that is, the least closed element which includes x. Besides, closure operators are in bijection with the associated sets of closed elements, known as closure systems or Moore families. In the case of complete lattices, every (finite or not) meet of closed elements is closed, so that every Moore family is itself a complete lattice, but not a sublattice of the original one. Level C is the framework of algebraic closures, whose associated closure systems describe an algebraic structure, or equivalently, a deductive structure. We have incorporated this equivalence and two intrinsic characterizations of algebraic closures, in a theorem which is emblematic for the unifying power of closure operators. The last part of the paper is devoted to compactly generated complete lattices, known as algebraic lattices. They are related to level C by a representation theorem. For properties of closure operators not included in this paper see e.g. [2, §§ 4.4, 4.5].