Coverages Give Free Constructions for Partial Frames

摘要

Defining objects using generators and relations has seen substantial application in the theory of frames. It is the aim of this paper to establish such a technique for partial frames, thus making it available in a variety of contexts. A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets in question are specified by means of a so-called selection function. The theory is general enough to include, as examples, bounded distributive lattices, sigma-frames, kappa-frames and indeed frames, but a small collection of elementary axioms suffices to describe the selection functions and thus the designated subsets. In this paper we are concerned with establishing techniques for constructing objects given certain generators and the relations that they should satisfy. Our method involves embedding the generators in an appropriate meet-semilattice, moving to the free partial frame over that meet-semilattice, and then using the relations to form a quotient with the required joins. We use a modification of Johnstone's coverages on meet-semilattices 12 to construct partial frames freely generated by sites. We conclude with a number of applications, including the construction of coproducts for partial frames and a general method for freely adjoining complements.