On covered prime elements and complete homomorphisms of frames

Abstract

It is a standard fact that the right adjoints h_* of general frame homomor- phisms h : L → M send prime elements to prime elements. Unlike this, the preservation of covered prime elements by complete frame homomorphism is a special fact. Indeed, if L is totally ordered this happens (for arbitrary M) if and only if L is well-ordered. On the other hand, if M is both a frame and a co-frame there is no restriction on L.

This work was motivated by the fact that, in an earlier paper, we erroneously claimed h_* preserves covered primeness for any complete frame homomorphism h.