Generalized Priestley Quasi-Orders

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 quasiorders (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 wellknown 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). 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. G. Bezhanishvili (B) Department of Mathematical Sciences, New Mexico State University, Las Cruces NM 88003-8001, USA e-mail: [email protected] R. Jansana Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001 Barcelona, Spain e-mail: [email protected] 202 Order (2011) 28:201–220