coframe

A coframe is a concept in order theory that is dual to a frame. Formally, a coframe is a complete lattice L in which finite joins distribute over arbitrary meets. Equivalently, for every element x in L and every family {y_i} in L, x ∨ (inf_i y_i) = inf_i (x ∨ y_i). Dually, a frame is a complete lattice where finite meets distribute over arbitrary joins.

Morphism and category: A map f between coframes is a coframe homomorphism if it preserves finite joins and arbitrary meets. In particular, it preserves binary joins and arbitrary meets of sets. The category of coframes with these morphisms is equivalent to the opposite category of frames; that is, Coframe is naturally equivalent to Frame^op.

Examples: The power set lattice P(X), ordered by inclusion, is a coframe because A ∪ (∩_i B_i) = ∩_i (A ∪ B_i) for any family {B_i}. Any complete Boolean algebra is both a frame and a coframe, since joins and meets distribute in all directions. The lattice of open sets of a topological space forms a frame; its dual perspective as a coframe arises by reversing the order.

Relation to other concepts: Coframes are the dual notion to frames, which model spaces in locale theory and point-free topology. Through duality, many results about frames have corresponding coframe versions. The study of coframes thus complements the theory of frames and locales and finds use in dualities between topological and algebraic structures, as well as in certain areas of domain theory.