Abstract
We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characterization of subalgebras of bounded distributive lattices by means of Priestley quasi-orders (Adams, Algebra Univers 3:216–228, 1973; Cignoli et al., Order 8(3):299–315, 1991; Schmid, Order 19(1):11–34, 2002). We also introduce Vietoris families and prove that homomorphic images of bounded distributive meet-semilattices are dually characterized by Vietoris families. We show that this generalizes the well-known characterization (Priestley, Proc Lond Math Soc 24(3):507–530, 1972) of homomorphic images of a bounded distributive lattice by means of closed subsets of its Priestley space. We also show how to modify the notions of generalized Priestley quasi-order and Vietoris family to obtain the dual characterizations of subalgebras and homomorphic images of bounded implicative semilattices, which generalize the well-known dual characterizations of subalgebras and homomorphic images of Heyting algebras (Esakia, Sov Math Dokl 15:147–151, 1974).
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Adams, M.E.: The Frattini sublattice of a distributive lattice. Algebra Univers. 3, 216–228 (1973)
Bezhanishvili, G., Jansana, R.: Duality for distributive and implicative semi-lattices. University of Barcelona research group in non-classical logics. http://www.mat.ub.edu/~logica/docs/BeJa08-m.pdf (2008, preprints)
Bezhanishvili, G., Jansana, R.: Esakia style duality for implicative semilattices (2010, submitted)
Bezhanishvili, G., Jansana, R.: Priestley style duality for distributive meet-semilattices (2010, submitted)
Cignoli, R., Lafalce, S., Petrovich, A.: Remarks on Priestley duality for distributive lattices. Order 8(3), 299–315 (1991)
Esakia, L.L.: Topological Kripke models. Sov. Math. Dokl. 15, 147–151 (1974)
Koppelberg, S.: Handbook of Boolean Algebras, vol. 1. North Holland, Amsterdam, The Netherlands (1989)
Nemitz, W.C.: Implicative semi-lattices. Trans. Am. Math. Soc. 117, 128–142 (1965)
Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)
Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. 24(3), 507–530 (1972)
Schmid, J.: Quasiorders and sublattices of distributive lattices. Order 19(1), 11–34 (2002)
Stone, M.H.: The theory of representations for Boolean algebras. Trans. Am. Math. Soc. 40(1), 37–111 (1936)
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The work of the first author was partially supported by the Georgian National Science Foundation grant GNSF/ST06/3-003.
The work of the second author was partially supported by 2009SGR-1433 research grant from the funding agency AGAUR of the Generalitat de Catalunya and by the MTM2008-01139 research grant of the Spanish Ministry of Education and Science.
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Bezhanishvili, G., Jansana, R. Generalized Priestley Quasi-Orders. Order 28, 201–220 (2011). https://doi.org/10.1007/s11083-010-9166-0
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DOI: https://doi.org/10.1007/s11083-010-9166-0