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Lattice uniformities and modular functions on orthomodular lattices. (English) Zbl 0834.06013
Traynor’s decomposition for group-valued set functions is generalized to exhaustive modular functions. It is shown that the lattice of exhaustive (lattice-) uniformities on an orthomodular lattice \(L\) is a complete Boolean algebra isomorphic to the centre of a quotient completion of \(L\).
Reviewer: G.Kalmbach (Ulm)

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
06F30 Ordered topological structures (aspects of ordered structures)
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[1] Ajupov, S. A., ?ilin, V. J., Chad?ijev, Z., and Sarymsakov, T. A. (1983)Ordered Algebras, FAN, Ta?kent.
[2] Drewnowski, L. (1972) Topological rings of sets, continuous set functions, integration I, II, III,Bull. Acad. Polon. Sci., Sér. Sci. Math. Astr. Phys. 20, 269-276, 277-286, 439-445. · Zbl 0249.28004
[3] Drewnowski, L. (1973) Decomposition of set functions,Studia Math. 48, 23-48. · Zbl 0269.28003
[4] Fleischer, I. and Traynor, T. (1982) Group-valued modular functions,Algebra Universalis 14, 287-291. · Zbl 0487.06002 · doi:10.1007/BF02483932
[5] Kalmbach, G. (1983)Orthomodular Lattices, Academic Press, London. · Zbl 0512.06011
[6] Palko, V. (1989) Topologies on quantum logics induced by measures,Math. Slovaca 39, 175-189. · Zbl 0674.03021
[7] Pták, P. and Pulmannova, S. (1991)Orthomodular Structures as Quantum Logics, Vol. 44, Kluwer Academic Publishers, London. · Zbl 0743.03039
[8] Pulmannova, S. and Rie?anova, Z. (1989) A topology on quantum logics,Proc. Amer. Math. Soc. 106, 891-897. · Zbl 0679.06004 · doi:10.2307/2047271
[9] Pulmannova, S. and Rie?anova, Z. (1991) Logics with separating sets of measures,Math. Slovaca 41, 167-177.
[10] Pulmannova, S. and Rie?anova, Z. (1991)Compact Topological Orthomodular Lattices, Contributions to general algebra7, 277-282; Teubner-Verlag, Stuttgart.
[11] Pulmannova, S. and Rogalewicz, V. (1991) Orthomodular lattices with almost orthogonal sets of atoms,Comment. Math. Univ. Carolinae 32, 423-429. · Zbl 0762.06003
[12] Rie?anova, Z. (1989) Topologies in atomic quantum logics,Acta Universitatis Carolinae ? Math. et Phys. 30, 143-148.
[13] Rie?anova, Z. (1988) Topology in a quantum logic induced by a measure, in Ch. Bandt, J. Flachsmeyer, and S. Lotz (eds),Proc. of the Conference Topology and Measure V, pp. 126-130.
[14] Traynor, T. (1976) The Lebesgue decomposition for group-valued set functions,Trans. Amer. Math. Soc. 220, 307-319. · Zbl 0334.28010 · doi:10.1090/S0002-9947-1976-0419725-8
[15] Weber, H. (1982) Unabhängige Topologien, Zerlegung von Ringtopologien,Math. Z. 180, 379-393. · Zbl 0485.13009 · doi:10.1007/BF01214178
[16] Weber, H. (1984) Topological Boolean rings. Decomposition of finitely additive set functions,Pac. J. Math. 110, 471-495. · Zbl 0489.28008
[17] Weber, H. (1984) Group- and vector-valueds-bounded contents, inMeasure Theory (Oberwolfach, 1983), Lecture Notes in Mathematics, Vol.1089, Springer-Verlag, pp. 181-198.
[18] Weber, H. (1991/1993) Uniform lattices I: A generalization of topological Riesz spaces and topological Boolean rings; Uniform lattices II: Order continuity and exhaustivity,Ann. Mat. Pura Appl. 160, 347-370;164, 133-158. · Zbl 0790.06006 · doi:10.1007/BF01764134
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