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)


06D15 Pseudocomplemented lattices
Full Text: DOI


[1] DOI: 10.1007/BF02485824 · Zbl 0353.06002
[2] Balbes, Distributive Lattices (1974)
[3] Adams, Czechoslovak Math. J. 41 pp 216– (1991)
[4] DOI: 10.1007/BF02488020 · Zbl 0402.06003
[5] Beazer, Studia Sci. Math.Hungar 20 pp 43– (1985)
[6] DOI: 10.1007/BF02485737 · Zbl 0316.06007
[7] DOI: 10.1007/BF02485372 · Zbl 0381.06019
[8] Burris, A course in universal algebra (1981) · Zbl 0478.08001
[9] DOI: 10.1016/0012-365X(90)90068-S · Zbl 0702.06011
[10] Grätzer, General lattice theory (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.