Extending semilattices to frames using sites and coverages

Abstract

Each meet semilattice S is well known to be freely extended to a frame by its down-sets DS. In this article we present, first, the complete range of frame extensions generated by S; it turns out to be a sub-coframe of the coframe C of sublocales of DS, indeed, an interval in C, with DS as the top and the extension of S respecting all the exact joins in S as the bottom.

Then, the Heyting and Boolean case is discussed; there, the bottom extension is shown to coincide with the Dedekind-MacNeille completion. The technique used is a technique of sites, generalizing that used in [JOHNSTONE, P.

T.: Stone Spaces. Cambridge Stud.

Adv. Math. 3, Cambridge University Press, Cambridge, 1982]. (C) 2014 Mathematical Institute Slovak Academy of Sciences

Keywords

• site
• coverage
• frame
• meet semilattice
• joins in meet semilattices