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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)

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)
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