| Abstract: | The cuts of the classical Dedekind-MacNeille completion DM(S) of a meet semilattice S give rise to a natural cut coverage in the down-set frame ({mathcal{D}S}): down-set D covers element s if s lies below all upper bounds of D. This, in turn, leads to what we call the Dedekind-MacNeille frame extension DMF(S). The meet semilattices S for which DM(S) = DMF(S), which we refer to as proHeyting semilattices, can be specified by a simple formula, and we provide a number of equivalent characterizations. A sample result is that DM(S) = DMF(S) iff DM(S) is a Heyting algebra iff DM(S) coincides with the Bruns-Lakser injective envelope. |
