A look at the structure of congruence frames by means of Heyting congruences

Abstract

This paper concerns the congruence frame in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, o and k frames. A new class of congruences, called Heyting congruences, helps to illuminate the structure of these congruence frames. Defining these involves the very useful free frame over a partial frame. We investigate the relationship between Heyting congruences and co-atoms of the congruence frame, showing how different the cases of full and partial frames can be. The Madden congruence, which yields the least dense quotient, is recognized as a particular Heyting congruence. While the collection of all Heyting congruences on a partial frame is in general not even a meet-semilattice, joining each of them with the Madden congruence results in a collection with considerably more interesting structure.