On pseudo-effect algebras which can be covered by pseudo MV-algebras

• Autorzy: Dvurečenskij A. , Vetterlein T.
• Języki publikacji: EN

Abstrakty

EN

Pseudo-effect algebras are partial algebras (E; +, 0,1) which were recently introduced. They have a partially defined addition + which is only associative and not necessary commutative and with two complements, left and right ones. They are a non-commutative generalization of orthomodular posets and MV-algebras, respectively. We define five kinds of compatibilities, and we introduce a block as a maximal set of mutually compatible elements. The compatibility is a property of the physical system which corresponds to the distributivity, or equivalently, to "classical mechanics"-type phenomena. We show that any lattice pseudo-effect algebra under a natural condition can be covered by blocks, and any block is a pseudo MV-algebra. This result generalizes the analogical result of Riecanova for effect algebras. If the pseudo-effect algebra with the condition is, in addition, a (7-complete lattice, then it is a commutative effect algebra which can be covered by cr-complete MV-algebras.

Słowa kluczowe

EN

MV-algebra; strong compatibility; effect algebras; pseudo-effect algebra

PL

MV-algebra

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2003
• Tom: Vol. 36, nr 2
• Strony: 261--282
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Dvurečenskij A.

• Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia

autor

Vetterlein T.

• Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia

Bibliografia

• [Bau] R. Baudot, Non-commutative logic programming language NoClog, in: Symposium LICS, Santa Barbara, 2000, Short Presentation, pp. 3-9.
• [Cha] C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467-490.
• [CGP] G. Cattaneo, R. Giuntini, S. Pulmannová, Pre-BZ and degenerate BZ posets: Applications to fuzzy sets and unsharp quantum theories, Found. Phys. 30 (2000), 1765-1799.
• [DGI] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL-algebras, I, II, Multi. Val. Logic 8 (2002), 673-714, 717-750.
• [Dvu] A. Dvurečenskij, Pseudo MV-algebras are intervals in I-groups, J. Austral. Math. Soc., 72 (2002), 427-445.
• [Dvu 1] A. Dvurečenskij, On effect algebras which can be covered by MV-algebras, Inter. J. Theor. Phys. 41 (2002), 221-229.
• [DvPu] A. Dvurečenskij, S. Pulmannová, New Trends in Quantum Structures, Kluwer Academic Publ., Dordrecht, 2000.
• [DvVe I] A. Dvurečenskij, T. Vetterlein, Pseudoeffect algebras. I. Basic properties, Inter. J. Theor. Phys. 40 (2001), 685-701.
• [DvVe II] A. Dvurečenskij, T. Vetterlein, Pseudoeffect algebras. II. Group representations, Inter. J. Theor. Phys. 40 (2001), 703-726.
• [Fou] D. J. Foulis, Sequential probability models and transition probabilities, Atti Semin. Mat. Fis. Univ. Modena, to appear.
• [FoBe] D. J. Foulis, M. K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325-1346.
• [Fuc] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, London, NY, Paris, 1963).
• [GeIo] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras, Multi. Val. Logic 6 (2001), 95-135.
• [GuNa] S. Gudder, G. Nagy, Sequentially independent effects, Proc. Amer. Math. Soc. 130 (2001) 1125-1130.
• [Jen] G. Jenča, Blocks of homogeneous effect algebras, Bull. Austral. Math. Soc., 64 (2001), 81-98.
• [Rac] J. Rachůnek, A non-commutative generalization of MV-algebras, Czechoslovak Math. J. 52 (2002), 255-273.
• [Rac 1] J. Rachůnek, Prime ideals and polars in generalized MV-algebras, Algebra Univer. 48 (2002), 151-169.
• [Rie] Z. Riečanová, A generalization of blocks for lattice effect algebras, Inter. J. Theoret. Phys. 39 (2000), 231-237.
• [Rie 1] Z. Riečanová, Mac Neille completion of D-posets and effect algebras, Inter. J. Theoret. Phys. 39 (2000), 859-869.
• [Rig] L. Rieger, On the ordered and cyclically ordered groups I, II, III, Vĕst. Král. České Spol. Nauk (1946, 1947, 1948) (in Czech).
• [Var] V.S. Varadarajan, Geometry of Quantum Theory, I, van Nostrand, Princeton, New Jersey, 1968.