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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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References:
 [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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