×
Adams, M. E.; Beazer, R.

Congruence properties of distributive double \(p\)-algebras. (English) Zbl 0758.06008

Czech. Math. J. 41(116), No. 2, 216-231 (1991).
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.
Reviewer: T.Katriňák (Bratislava)

Cited in 1 Review
Cited in 5 Documents

MSC:

06D15 Pseudocomplemented lattices
08A30 Subalgebras, congruence relations
06B10 Lattice ideals, congruence relations

Keywords:

congruence lattice; distributive double \(p\)-algebra; congruence permutable; relatively complemented; prime ideals; principal join property; principal congruences
PDFBibTeX XMLCite
\textit{M. E. Adams} and \textit{R. Beazer}, Czech. Math. J. 41(116), No. 2, 216--231 (1991; Zbl 0758.06008)
Full Text: DOI EuDML

References:

[1] R. Beazer: The determination congruence on double \(p\)-algebras. Algebra Universalis 6 (1976), 121-129. · Zbl 0353.06002 · doi:10.1007/BF02485824
[2] R. Beazer: Pseudocomplemented algebras with Boolean congruence lattices. J. Australian Math. Soc. 26 (1978), 163-168. · Zbl 0389.06004 · doi:10.1017/S1446788700011654
[3] R. Beazer: Congruence pairs of distributive doubles-algebras with non-empty core. Houston J. Math. 6 (1980), 443-454. · Zbl 0464.06006
[4] R. Beazer: Congruence uniform algebras with pseudocomplementation. Studia Sci. Math. Hungar. 20 (1985), 43-48 · Zbl 0523.06016
[5] S. Burris, H. P. Sankappanavar: A Course in Universal Algebra. Graduate Texts in Mathematics, Springer-Verlag, 1981. · Zbl 0478.08001
[6] I. Chajda: A Maľcev condition for congruence principal permutable varieties. Algebra Universalis 19 (1984), 337-340. · Zbl 0552.08006 · doi:10.1007/BF01201102
[7] G. Grätzer: General Lattice Theory. Birkhäuser Verlag, Basel and Stuttgart, 1978. · Zbl 0385.06015
[8] T. Hecht, T. Katriňák: Principal congruences of \(p\)-algebras and double \(p\)-algebras. Proc. Amer. Math. Soc. 58 (1976), 25-31. · Zbl 0352.06006
[9] T. Katriňák: The structure of distributive double \(p\)-algebras. Regularity and congruences. Algebra Universalis 3 (1973), 238-246. · Zbl 0276.08005 · doi:10.1007/BF02945123
[10] T. Katriňák: Congruence extension property for distributive double \(p\)-algebras. Algebra Universalis 4 (1974), 273-276. · Zbl 0316.06007 · doi:10.1007/BF02485737
[11] T. Katriňák: Congruence pairs on \(p\)-algebras with a modular frame. Algebra Universalis 8 (1978), 205-220. · Zbl 0381.06017 · doi:10.1007/BF02485390
[12] T. Katriňák: Subdirectly irreducible distributive double \(p\)-algebras. Algebra Universalis 10 (1980), 195-219. · Zbl 0431.06013 · doi:10.1007/BF02482902
[13] V. Koubek, J. Sichler: Universal varieties of distributive double \(p\)-algebras. Glasgow Math. J. 26 (1985), 121-133. · Zbl 0574.06009 · doi:10.1017/S0017089500005887
[14] R. W. Quackenbush: Varieties with \(n\)-principal compact congruences. Algebra Universalis 14 (1982), 292-296. · Zbl 0493.08006 · doi:10.1007/BF02483933
[15] J. C. Varlet: A regular variety of type . Algebra Universalis 2(1972), 218-223. · Zbl 0256.06004 · doi:10.1007/BF02945029
[16] J. C. Varlet: Large congruences in \(p\)-algebras and double \(p\)-algebras. Algebra Universalis 9 (1979), 165-178. · Zbl 0436.06009 · doi:10.1007/BF02488028
[17] J. C. Varlet: Regularity in double \(p\)-algebras. Algebra Universalis 18 (1984), 95-105. · Zbl 0547.06008 · doi:10.1007/BF01182250
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.