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Commutator-finite D-lattices. (English) Zbl 1081.06013
The author studies lattice-ordered D-posets (= D-lattices). She finds conditions when a commutator-finite D-lattice $$L$$ can be uniquely decomposed in the form $$L \cong M\times L_1 \times \cdots\times L_n$$, where $$M$$ is an MV-algebra and $$L_1,\ldots,L_n$$ are irreducible D-lattices which are not MV-algebras. The basic problem in question is the presence of so-called unsharp elements. In addition, she proves that the class of commutator-finite D-lattices contains, under some conditions, the class of block-finite D-lattices. This allows her to study some interesting questions concerning states and valuations.

##### MSC:
 06D35 MV-algebras 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G12 Quantum logic
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##### References:
 [1] Avallone, A. and Vitolo, P.: Congruences and ideals of effect algebras, Order 20 (2003), 67–77. · Zbl 1030.03047 [2] Bennett, M. K. and Foulis, D. J.: Phi-symmetric effect algebras, Found. Phys. 25 (1995), 1699–1722. [3] Beran, L.: Orthomodular Lattices. Algebraic Approach, Academia Prague–D. Reidel, Dordrecht, 1984. [4] Bruns, G.: Block-finite orthomodular lattices, Canad. J. Math. 31 (1979), 961–985. · Zbl 0429.06002 [5] Bruns, G. and Greechie, R.: Some finiteness conditions for orthomodular lattices, Canad. J. Math. 34 (1982), 535–549. · Zbl 0494.06008 [6] Bruns, G. and Greechie, R.: Orthomodular lattices which can be covered by finitely many blocks, Canad. J. Math. 34 (1982), 696–699. · Zbl 0493.06008 [7] Bruns, G. and Greechie, R.: Blocks and commutators in orthomodular lattices, Algebra Universalis 27 (1990), 1–9. · Zbl 0694.06006 [8] Chang, C. C.: Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc. 88 (1958), 467–490. · Zbl 0084.00704 [9] Chevalier, G. and Pulmannová, S.: Some ideal lattices in partial Abelian monoids and effect algebras, Order 17 (2000), 75–92. · Zbl 0960.03053 [10] Chovanec, F. and Kôpka, F.: D-lattices, Internat. J. Theor. Phys. 34 (1995), 1297–1302. · Zbl 0840.03046 [11] Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D.: Algebraic Foundations of Many-valued Reasoning, Kluwer Academic Publ., Dordrecht, 2000. · Zbl 0937.06009 [12] Dvurečenskij, A. and Pulmannová, S.: New Trends in Quantum Structures, Kluwer Academic Publ./Ister Science Press, Dordrecht/Bratislava, 2000. · Zbl 0987.81005 [13] Foulis, D. and Bennett, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331–1352. · Zbl 1213.06004 [14] Giuntini, R. and Greuling, H.: Toward a formal language for unsharp properties, Found. Phys. 19 (1994), 769–780. [15] Greechie, R. and Herman, L.: Commutator-finite orthomodular lattices, Order 1 (1985), 277–284. · Zbl 0553.06012 [16] Grätzer, G.: Universal Algebra, 2nd edn, Springer-Verlag, 1979, pp. 80–81. [17] Greechie, R., Foulis, D. and Pulmannová, S.: The center of an effect algebra, Order 12 (1995), 91–106. · Zbl 0846.03031 [18] Gudder, S. and Pulmannová, S.: Quotients of partial Abelian monoids, Algebra Universalis 38 (1997), 395–421. · Zbl 0933.03082 [19] Jenča, G. and Pulmannová, S.: Ideals and quotients in lattice ordered effect algebras, Soft Computing 5 (2001), 376–380. · Zbl 1004.06009 [20] Jenča, G. and Pulmannová, S.: Orthocomplete effect algebras, Proc. Amer. Math. Soc. 131 (2003), 2663–2671. · Zbl 1019.03046 [21] Jenča, G. and Riečanová, Z.: On sharp elements in lattice ordered effect algebras, Busefal 80 (1999), 24–29. [22] Kalmbach, G.: Orthomodular Lattices, Academic Press, London, 1983. · Zbl 0512.06011 [23] Kôpka, F. and Chovanec, F.: D-posets, Math. Slovaca 44 (1994), 21–34. [24] Maeda, F. and Maeda, S.: Theory of Symmetric Lattices, Springer-Verlag, Berlin, 1970. · Zbl 0219.06002 [25] Marsden, E.: The commutator and solvability in generalized orthomodular lattices, Pacific J. Math. 33 (1970), 357–361. · Zbl 0234.06004 [26] Pulmannová, S.: Congruences in partial Abelian semigroups, Algebra Universalis 37 (1997), 119–140. · Zbl 0902.08008 [27] Riečanova, Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras, Internat. J. Theor. Phys. 39 (2000), 231–237. · Zbl 0968.81003 [28] Riečanová, Z.: Block-finite effect algebras and the existence of states, Demonstratio Math. 36 (2003), 507–517. · Zbl 1042.03041
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