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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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