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The principal join property in demi-p-lattices. (English) Zbl 1150.06010
A demi-p-lattice is a bounded distributive lattice equipped with an additional unary operation, satisfying certain identities. Demi-p-lattices form a subvariety of the variety (equational class) of semi-De Morgan algebras.
An algebra $$L$$ has the principal join property (PJP) if the join of any two principal congruences is again a principal congruence. The principal intersection property (PIP) is defined similarly.
The main result of this paper characterizes the demi-p-lattices with PJP, both algebraically and in terms of the dual Priestley spaces. An additional characterization is given for some special demi-p-lattices, called almost-p-lattices. Finally, it is proved that, in demi-p-lattices, PJP implies PIP.
##### MSC:
 06D15 Pseudocomplemented lattices
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##### References:
 [1] ADAMS M. E.-BEAZER R.: Congruence properties of distributive double p-algebras. Czechoslvak Math. J. 41 (1991), 216-231. · Zbl 0758.06008 [2] ADAMS M. E.: Principal congruences in De Morgan algebras. Proc. Edinburgh Math. Soc. (2) 30 (1987), 415-421. · Zbl 0595.06013 [3] BALBES R.-DWINGER P.: Distributive Lattices. University of Missouri Press, Columbia, MO, 1974. · Zbl 0321.06012 [4] BEAZER R.: Principal congruence properties of some algebras with pseudocomplementation. Portugal. Math. 50 (1993), 75-86. · Zbl 0801.06023 [5] BEAZER R.: Some p-algebras and double p-algebras having only principal congruences. Glasg. Math. J. 34 (1992), 157-164. · Zbl 0776.06008 [6] BLYTH T. S.-VARLET J. C.: Principal congruences on some lattice-ordered algebras. Discrete Math. 81 (1990), 323-329. · Zbl 0702.06011 [7] CHAJDA I.: A Maľcev condition for congruence principal permutable varieties. Algebra Universalis 19 (1984), 337-340. · Zbl 0552.08006 [8] CHAJDA I.: Algebras whose principal congruences form a sublattice of the congruence lattice. Czechoslovak Math. J. 38(113) (1988), 585-588. · Zbl 0668.08003 [9] DAVEY B. A.-PRIESTLEY H. A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge, 1990. · Zbl 0701.06001 [10] GRÄTZER G.: General Lattice Theory. Mathematische Reihe Bd. 52. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Birkhäuser Verlag, Basel-Stuttgart, 1978. · Zbl 0436.06001 [11] HOBBY D.: Semi-De Morgan algebras. Studia Logica 56 (1996), 150-183. · Zbl 0854.06010 [12] PALMA C.-SANTOS R.: On the subdirectly irreducible semi-De Morgan algebras. Publ. Math. Debrecen 49 (1996), 39-45. · Zbl 0868.06006 [13] PALMA C.-SANTOS R.: Principal congruences on Semi-De Morgan Algebras. Studia Logica67 (2001), 75-88. · Zbl 0981.06007 [14] PRIESTLEY H. A.: The construction of spaces dual to pseudocomplemented distributive lattices. Quart. J. Math. Oxford Ser. (2) 26 (1975), 215-228. · Zbl 0323.06013 [15] SANKAPPANAVAR H. P.: Semi-De Morgan algebras. J. Symbolic Logic 52 (1987), 712-724. · Zbl 0628.06011 [16] SANKAPPANAVAR H. P.: Demi-pseudocomplemented lattices: principal congruences and subdirectly irreducibility. Algebra Universalis 27 (1990), 180-193. · Zbl 0701.06009 [17] SANKAPPANAVAR H. P.: Varieties of demi-pseudocomplemented lattices. Z. Math. Logik Grundlag. Math. 37 (1991), 411-420. · Zbl 0769.06007
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