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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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