framesubset

framesubset

In frame theory, a framesubset (often simply called a subframe) of a frame L is a subset S of L that is itself a frame with the induced order and operations. Concretely, S must contain the bottom and top elements (0 and 1) of L, and be closed under finite meets (finite intersections) and arbitrary joins (unions). Under these conditions, S, with the inherited order, join, and meet, is a frame in its own right.

A framesubset S is equivalent to a frame embedding i: S → L, where the inclusion map preserves finite meets and arbitrary joins. This makes S a substructure of L that reflects the lattice-theoretic and distributive properties of frames. Not every subset of a frame is a framesubset; the closure properties under meets and joins are essential.

Examples help illustrate the concept. In the power set frame P(X) with union as join and intersection as meet, framesubsets correspond to complete Boolean subalgebras of P(X) containing both the empty set and X. Such subframes can be described as collections of sets closed under arbitrary unions and finite intersections that include 0 and 1. More generally, subframes often arise by restricting a frame to a chosen family of elements closed under the required operations.

Framesubsets are relevant in the study of locale theory and point-free topology, where frames model spaces and subframes model subspaces in a lattice-theoretic sense. They provide a way to examine internal substructure while preserving the frame properties that enable the duality with topological or geometric notions.