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Congruence properties of distributive double $$p$$-algebras. (English) Zbl 0758.06008
An algebra $$(L;\lor,\land,^*,^ +,0,1)$$ of type (2,2,1,1,0,0) is called a (distributive) double $$p$$-algebra if $$(L;\lor,\land,0,1)$$ is a bounded (distributive) lattice in which, for any $$a\in L$$, $$a^*$$ (= the pseudocomplement of $$a$$) is characterized by $$a\land x=0$$ if and only if $$x\leq a^*$$, and $$a^ +$$ (= the dual pseudocomplement of $$a$$) is defined dually.
The main results: Let $$L$$ be a distributive double $$p$$-algebra. (1) The following conditions are equivalent: (i) $$L$$ is congruence permutable; (ii) the sublattices $$\{x\in L: x^*=a^*$$ and $$x^ +=a^ +\}$$, $$s\in L$$, are relatively complemented; (iii) the poset of prime ideals of $$L$$ contains no 4-element chain. (2) The following conditions are equivalent: (i) $$L$$ enjoys the P.J.P. (= principal join property, i.e. the join of any pair of principal congruences on $$L$$ is again principal); (ii) there is no 3-element chain in the poset of prime ideals of $$L$$; (iii) the sublattices $$\{x\in L: x^*=a^*$$ and $$x^ +=a^ +\}$$, $$a\in L$$, enjoy the P.J.P.

##### MSC:
 06D15 Pseudocomplemented lattices 08A30 Subalgebras, congruence relations 06B10 Lattice ideals, congruence relations
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##### References:
 [1] R. Beazer: The determination congruence on double $$p$$-algebras. Algebra Universalis 6 (1976), 121-129. · Zbl 0353.06002 [2] R. Beazer: Pseudocomplemented algebras with Boolean congruence lattices. J. Australian Math. Soc. 26 (1978), 163-168. · Zbl 0389.06004 [3] R. Beazer: Congruence pairs of distributive doubles-algebras with non-empty core. Houston J. Math. 6 (1980), 443-454. · Zbl 0464.06006 [4] R. Beazer: Congruence uniform algebras with pseudocomplementation. Studia Sci. Math. Hungar. 20 (1985), 43-48 · Zbl 0523.06016 [5] S. Burris, H. P. Sankappanavar: A Course in Universal Algebra. Graduate Texts in Mathematics, Springer-Verlag, 1981. · Zbl 0478.08001 [6] I. Chajda: A Maľcev condition for congruence principal permutable varieties. Algebra Universalis 19 (1984), 337-340. · Zbl 0552.08006 [7] G. Grätzer: General Lattice Theory. Birkhäuser Verlag, Basel and Stuttgart, 1978. · Zbl 0385.06015 [8] T. Hecht, T. Katriňák: Principal congruences of $$p$$-algebras and double $$p$$-algebras. Proc. Amer. Math. Soc. 58 (1976), 25-31. · Zbl 0352.06006 [9] T. Katriňák: The structure of distributive double $$p$$-algebras. Regularity and congruences. Algebra Universalis 3 (1973), 238-246. · Zbl 0276.08005 [10] T. Katriňák: Congruence extension property for distributive double $$p$$-algebras. Algebra Universalis 4 (1974), 273-276. · Zbl 0316.06007 [11] T. Katriňák: Congruence pairs on $$p$$-algebras with a modular frame. Algebra Universalis 8 (1978), 205-220. · Zbl 0381.06017 [12] T. Katriňák: Subdirectly irreducible distributive double $$p$$-algebras. Algebra Universalis 10 (1980), 195-219. · Zbl 0431.06013 [13] V. Koubek, J. Sichler: Universal varieties of distributive double $$p$$-algebras. Glasgow Math. J. 26 (1985), 121-133. · Zbl 0574.06009 [14] R. W. Quackenbush: Varieties with $$n$$-principal compact congruences. Algebra Universalis 14 (1982), 292-296. · Zbl 0493.08006 [15] J. C. Varlet: A regular variety of type . Algebra Universalis 2(1972), 218-223. · Zbl 0256.06004 [16] J. C. Varlet: Large congruences in $$p$$-algebras and double $$p$$-algebras. Algebra Universalis 9 (1979), 165-178. · Zbl 0436.06009 [17] J. C. Varlet: Regularity in double $$p$$-algebras. Algebra Universalis 18 (1984), 95-105. · Zbl 0547.06008
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