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Decompositions for relatively normal lattices. (English) Zbl 0799.06019
A lower-bounded distributive lattice is called relatively normal if in its set of prime ideals \(P\) ordered by set-inclusion every principal upper set is a chain. The most general conditions are obtained under which a relatively normal lattice may be represented as a union of its special ideals (Theorem B). It is also shown that if for a lower-bounded distributive lattice \(L\) its quotient lattice \(L/\theta\) relative to the Glivenko congruence \(\theta\) satisfies the descending chain condition, then \(L\) is relatively normal iff \(L\) is isomorphic to the lattice of all principal convex \(\ell\)-subgroups of an abelian \(\ell\)-group (Theorem D).

MSC:
06D05 Structure and representation theory of distributive lattices
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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[1] Marlow Anderson, Paul Conrad, and Jorge Martinez, The lattice of convex \?-subgroups of a lattice-ordered group, Lattice-ordered groups, Math. Appl., vol. 48, Kluwer Acad. Publ., Dordrecht, 1989, pp. 105 – 127. · doi:10.1007/978-94-009-2283-9_6 · doi.org
[2] Marlow Anderson and Todd Feil, Lattice-ordered groups, Reidel Texts in the Mathematical Sciences, D. Reidel Publishing Co., Dordrecht, 1988. An introduction. · Zbl 0636.06008
[3] Raymond Balbes and Philip Dwinger, Distributive lattices, University of Missouri Press, Columbia, Mo., 1974. · Zbl 0321.06012
[4] R. Beazer, Hierarchies of distributive lattices satisfying annihilator conditions, J. London Mat. Soc. (2) 11 (1975), no. 2, 216 – 222. · Zbl 0335.06008
[5] Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). · Zbl 0384.06022
[6] Gabriela Bordalo, Strongly \?-normal lattices, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 2, 113 – 125. · Zbl 0547.06004
[7] G. Bordalo and H. A. Priestley, Relative Ockham lattices: their order-theoretic and algebraic characterisation, Glasgow Math. J. 32 (1990), no. 1, 47 – 66. · Zbl 0693.06009 · doi:10.1017/S0017089500009058 · doi.org
[8] Paul Conrad, The structure of a lattice-ordered group with a finite number of disjoint elements, Michigan Math. J 7 (1960), 171 – 180. · Zbl 0103.01501
[9] Paul Conrad, Some structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc. 99 (1961), 212 – 240. · Zbl 0099.25401
[10] Paul Conrad, The lattice of all convex \?-subgroups of a lattice-ordered group, Czechoslovak Math. J. 15 (90) (1965), 101 – 123 (English, with Russian summary). · Zbl 0135.06301
[11] Paul Conrad, Lex-subgroups of lattice-ordered groups, Czechoslovak Math. J. 18 (93) (1968), 86 – 103 (English, with Loose Russian summary). · Zbl 0155.05902
[12] -, Lattice-ordered groups, Tulane Univ. lecture notes, New Orleans, 1970. · Zbl 0258.06011
[13] William H. Cornish, Normal lattices, J. Austral. Math. Soc. 14 (1972), 200 – 215. · Zbl 0247.06009
[14] William H. Cornish, \?-normal lattices, Proc. Amer. Math. Soc. 45 (1974), 48 – 54. · Zbl 0294.06008
[15] P. Crawley and R. P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Englewoods Cliffs, NJ, 1973. · Zbl 0494.06001
[16] Brian A. Davey, Some annihilator conditions on distributive lattices, Algebra Universalis 4 (1974), 316 – 322. · Zbl 0299.06007 · doi:10.1007/BF02485743 · doi.org
[17] George Grätzer, General lattice theory, Birkhäuser Verlag, Basel-Stuttgart, 1978. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52. George Grätzer, General lattice theory, Pure and Applied Mathematics, vol. 75, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.
[18] P. Jaffard, Contribution a l’etude des groupes ordonees, J. Math. Pures Appl. 32 (1953), 208-280.
[19] Peter T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. · Zbl 0499.54001
[20] Mark Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970), 377 – 386. · Zbl 0206.29701
[21] G. Markowsky and B. K. Rosen, Bases for chain-complete posets, IBM J. Res. Develop. 20 (1976), no. 2, 138 – 147. · Zbl 0329.06001 · doi:10.1147/rd.202.0138 · doi.org
[22] D. B. McAlister, On multilattice groups. II, Proc. Cambridge Philos. Soc. 62 (1966), 149 – 164. · Zbl 0138.02702
[23] R. McKenzie, G. McNulty, and W. Taylor, Algebras, lattices, varieties, vol. 1, Wadsworth and Brooks/Cole, Monterey, CA, 1987. · Zbl 0611.08001
[24] António A. Monteiro, L’arithmétique des filtres et les espaces topologiques, Segundo symposium sobre algunos problemas matemáticos que se están estudiando en Latino América, Julio, 1954, Centro de Cooperación Cientifica de la UNESCO para América Latina, Montevideo, Uruguay, 1954, pp. 129 – 162 (French). · Zbl 0058.38503
[25] -, L’arithmetique des filtres et les espaces topologiques. I, II, Notas Lógica Mat. (1974), 29-30.
[26] J. T. Snodgrass and C. Tsinakis, Finite-valued algebraic lattices, Algebra Universalis 30 (1993), no. 3, 311 – 318. · Zbl 0806.06011 · doi:10.1007/BF01190439 · doi.org
[27] J. T. Snodgrass and C. Tsinakis, The finite basis theorem for relatively normal lattices, Algebra Universalis 33 (1995), no. 1, 40 – 67. · Zbl 0819.06009 · doi:10.1007/BF01190765 · doi.org
[28] A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. · Zbl 0519.46001
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