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.
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Presented by P. P. Palfy.
In memory of Jiří Sichler
The first two authors gratefully acknowledge support from Project P202/12/G061 of the Grant Agency of the Czech Republic, and from the Department of Mathematics of the University of Denver. All three authors gratefully acknowledge support from the Center of Excellence in Applied Computational and Fundamental Science, Chapman University, Orange, California.
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Ball, R.N., Pultr, A. & Walters Wayland, J. The Dedekind MacNeille site completion of a meet semilattice. Algebra Univers. 76, 183–197 (2016). https://doi.org/10.1007/s00012-016-0397-9
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DOI: https://doi.org/10.1007/s00012-016-0397-9