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Some \(p\)-algebras and double \(p\)-algebras having only principal congruences. (English) Zbl 0776.06008
The author shows that a quasi-modular \(p\)-algebra \(L\) has only principal congruences if and only if the Boolean algebra \(B(L)=\{x\in L\mid x=x^{**}\}\) is finite and the filter \(D^*(L)=\{x\in L\mid x^{**}=1\}\) has only principal congruences. Although there are infinite modular \(p\)-algebras having only principal congruences, he shows that a distributive \(p\)-algebra has only principal congruences if and only if it is finite and the length of its poset of non-zero join-irreducibles \(\leq 2\). Again there are infinite distributive double \(p\)-algebras having only principal congruences. The author considers the problem for some distributive double \(p\)-algebras.
Reviewer: C.S.Hoo (Edmonton)

MSC:
06D15 Pseudocomplemented lattices
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