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Congruences on distributive pseudocomplemented lattices

Published online by Cambridge University Press:  17 April 2009

William H. Cornish
William H. Cornish
Affiliation:
School of Mathematical Sciences, Flinders University of South Australia, Bedford Park, South Australia.
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Abstract

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Let B0 ⊂ B1 ⊂ Bn ⊂ … ⊂: Bw be all the non-trivial varieties of distributive pseudocomplemented lattices (L; ∨, ∧, *, 0, 1) considered as algebras of type (2, 2, 1, 0, 0). A subset J of such an algebra L is a congruence-kernel if and only if it is a lattice-ideal and x** ∈ J for each x ∈ J. The smallest congruence having J as its kernel is Θ(J), where a ≡ b (Θ(J)), (a, b ∈ L) if and only if a ∧ c* = b ∧ c* for some c ∈ J. For given 0 ≤ n ≤ w, let Σn (J) be the smallest congruence having J as its kernel and such that the associated quotient algebra is in Bn. Of course, Σw (J) = Θ(J) and the main result of this paper shows that for 1 ≤ n < w, Σn(J) = ∩{Θ(P1 ∩ P2 ∩ … ∩ Pn): J ⊆ P1, P2, …, Pn ⊂ L are minimal prime ideals}.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 2 , April 1973 , pp. 161 - 179
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Cornish, William H., “Normal lattices”, J. Austral. Math. Soc. 14 (1972), 200–215.CrossRefGoogle Scholar
[2]Cornish, William H., “Annulets and α-ideals in a distributive lattice”, J. Austral. Math. Soc. (to appear).Google Scholar
[3]Grätzer, George, Universal algebra (Van Nostrand, Princeton, New Jersey; Toronto; London; Melbourne; 1968).Google Scholar
[4]Grätzer, George, Lattice theory; first concepts and distributive lattices (Freeman, San Francisco, 1971).Google Scholar
[5]Grätzer, G. and Schmidt, E.T., “Standard ideals in lattices”, Acta Math. Acad. Sci. Hungar. 12 (1961), 17–86.CrossRefGoogle Scholar
[6]Iqbalunnisa, , “Permutable congruences in a lattice”, Illinois J. Math. 10 (1966), 235–239.CrossRefGoogle Scholar
[7]Kist, Joseph, “Minimal prime ideals in commutative semigroups”, Proc. London Math. Soc. (3) 13 (1963), 31–50.CrossRefGoogle Scholar
[8]Lee, K.B., “Equational classes of distributive pseudo-complemented lattices”, Canad. J. Math. 22 (1970), 881–891.CrossRefGoogle Scholar
[9]Speed, T.P., “Some remarks on a class of distributive lattices”, J. Austral. Math. Soc. 9 (1969), 289–296.CrossRefGoogle Scholar
[10]Speed, T.P., “Two congruences on distributive lattices”, Bull. Soc. Roy. Sci. Liège 38 (1969), 86–95.Google Scholar