Generalized Priestley quasi-orders
The authors investigate bounded distributive meet-semilattices, and especially bounded implicative meet-semilattices. First, they develop a new `Priestley-like' duality for the category of bounded distributive meet-semilattices with homomorphisms. Secondly, they extend their description to a duality between the lattice Sub\(_{(0,1)}(S)\) of all subalgebras of a distributive meet-semilattice \(S\) and the space of generalized Priestley quasi-orders GP\((S).\) The final result is a generalization of a characterization given by \textit{M. E. Adams} [Algebra Univers. 3, 216--228 (1973; Zbl 0288.06015)] for bounded distributive lattices. In addition, a similar description of bounded implicative semilattices is obtained, generalizing a former result by \textit{L. L. Esakia} [Sov. Math., Dokl. 15, 147--151 (1974); translation from Dokl. Akad. Nauk SSSR 214, 298--301 (1974; Zbl 0296.02030)] for Heyting algebras.
scientific article; zbMATH DE number 5939491