Stone style duality for distributive nearlattices. (English) Zbl 1301.06030

Celani, Sergio; Calomino, Ismael

Algebra Univers. 71, No. 2, 127-153 (2014).

Summary: The aim of this paper is to study the variety of distributive nearlattices with greatest element. We will define the class of \(N\)-spaces as sober-like topological spaces with a basis of open, compact, and dually compact subsets satisfying an additional condition. We will show that the category of distributive nearlattices with greatest element whose morphisms are semi-homomorphisms is dually equivalent to the category of \(N\)-spaces with certain relations, called \(N\)-relations. In particular, we give a duality for the category of distributive nearlattices with homomorphisms. Finally, we apply these results to characterize topologically the one-to-one and onto homomorphisms, the subalgebras, and the lattice of the congruences of a distributive nearlattice.

MSC:

06D50	Lattices and duality
06D75	Other generalizations of distributive lattices

Keywords:

distributive nearlattices; prime ideal; topological representation; Stone spaces

References:

[1]	Abbott, J.C., Semi-Boolean algebra, Mat. Vestnik, 19, 177-198, (1967) · Zbl 0153.02704
[2]	Araújo, R.; Kinyon, M., Independent axiom systems for nearlattices, Czech. Math. J., 61, 975-992, (2011) · Zbl 1249.06003
[3]	Balbes, R., Dwinger, P.H.: Distributive Lattices. University of Missouri Press (1974) · Zbl 1132.08002
[4]	Bezhanishvili, G.; Jansana, R., Priestley style duality for distributive meet-semilattices, Studia Logica, 98, 83-122, (2011) · Zbl 1238.03050
[5]	Celani, S.A., Topological representation of distributive semilattices, Scientiae Math. Japonicae, 8, 41-51, (2003) · Zbl 1041.06002
[6]	Celani, S.A.; Cabrer, L.M., Topological duality for Tarski algebras, Algebra Universalis, 58, 73-94, (2008) · Zbl 1136.03046
[7]	Chajda, I.; Halaš; R., An example of a congruence distributive variety having no near-unanimity term, Acta Univ. M. Belii Math., 13, 29-31, (2006) · Zbl 1132.08002
[8]	Chajda, I., Halaš, R., Kühr J.: Semilattice Structures. Research and Exposition in Mathematics, vol. 30. Heldermann (2007) · Zbl 0436.06003
[9]	Chajda, I., Kolařík, M.: Ideals, congruences and annihilators on nearlattice. Acta Univ. Palacki. Olomuc. Fac. rer. nat. Math. 46, 25-33 (2007) · Zbl 0153.02704
[10]	Chajda, I., Kolařík, M.: Nearlattices. Discrete Math. 308, 4906-4913 (2008) · Zbl 1151.06004
[11]	Cornish, W.H.; Hickman, R.C., Weakly distributive semilattices, Acta Math. Acad. Sci. Hungar., 32, 5-16, (1978) · Zbl 0497.06005
[12]	Grätzer, G.: General Lattice Theory. Birkhäuser (1998) · Zbl 0909.06002
[13]	Halaš, R.: Subdirectly irreducible distributive nearlattices. Miskolc Mathematical Notes 7, 141-146 (2006) · Zbl 1120.06003
[14]	Hickman, R.C., Join algebras, Comm. Algebra, 8, 1653-1685, (1980) · Zbl 0436.06003
[15]	Stone, M.: Topological representation of distributive lattices and Brouwerian logics. Casopis pešt. mat. fys. 67, 125 (1937)