Partial frames, their free frames and their congruence frames

TL;DR

This paper explores the structure of partial frames, their free frames, and congruence frames, introducing new concepts and characterizations for their properties, including Booleanness, and analyzing the relationships between different types of congruences.

Contribution

It introduces the notion of partial frames with designated subsets, studies their free and congruence frames, and distinguishes various Booleanness conditions in this context.

Findings

Characterization of partial frames and their axioms.

Analysis of congruence frames and their relation to free frames.

Conditions for isomorphism between congruence frames and free frames.

Abstract

The context of this work is that of partial frames; these are meet-semilattices where not all subsets need have joins. A selection function, S, specifies, for all meet-semilattices, certain subsets under consideration, which we call the ``designated'' ones; an S-frame then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, sigma-frames, kappa-frames and frames. We consider right and left adjoints of S-frame maps, as a prelude to the introduction of closed and open maps. Then we look at what might be an appropriate notion of Booleanness for partial frames. The obvious candidate is the condition that every element be complemented; this concept is indeed of interest, but we pose three further conditions which, in the frame setting, are all equivalent to it. However, in the context of partial frames, the four conditions are distinct. In investigating these, we make essential use of the free frame over a partial frame and the congruence frame of a partial frame. We compare congruences of a partial frame, technically called S-congruences, with the frame congruences of its free frame. We provide a natural transformation for the situation and also consider right adjoints of the frame maps in question. We characterize the case where the two congruence frames are isomorphic and provide examples which illuminate the possible different behaviour of the two. We conclude with a characterization of closedness and openness for the embedding of a partial frame into its free fame, and into its congruence frame.