Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T14:55:17.118Z Has data issue: false hasContentIssue false
  • English
  • Français

Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras

Part of: Algebraic logic General logic

Published online by Cambridge University Press:  09 April 2009

Anatolij Dvurečenskij
Anatolij Dvurečenskij
Affiliation:
Mathematical InstituteSlovak Academy of Sciences Štefánikova 49 SK-814 73 Bratislava Slovakia e-mail: dvurecen@mat.savba.sk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Pseudo-effect algebras are partial algebras (E; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.

Keywords

Pseudo-effect algebraeffect algebracentral elementgeneral comparabilitypseudo MV-algebramonotone σ-completenessCantor-Bernstein theorem

MSC classification

Secondary: 03G12: Quantum logic 03B50: Many-valued logic
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 1 , February 2003 , pp. 121 - 144
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Baudot, R., ‘Non-commutative logic programming language NoClog’, in: Symposium LICS (Santa Barbara, 2000, Short Presentation) pp. 3–9.Google Scholar
[2]Belluce, L. P., ‘α-complete MV-algebras’, in: Non-classical logics and their applications to fuzzy subsets (ed. Hoehle, U. et al. ) (Kluwer, Dordrecht, 1995) pp. 7–21.CrossRefGoogle Scholar
[3]Chang, C. C., ‘Algebraic analysis of many valued logics’, Trans. Amer. Math. Soc. 88 (1958), 467–490.CrossRefGoogle Scholar
[4]De Simone, A., Mundici, D. and Navara, M., ‘A Cantor-Bernstein theorem for σ-complete MValgebras’, Czechoslovak Math. J. (to appear).Google Scholar
[5]De Simone, A., Navara, M. and Pták, P., ‘On interval homogeneous orthomodular lattices’, Comm. Math. Univ. Carolinae 42 (2001), 23–30.Google Scholar
[6]Di Nola, A., Georgescu, G. and Iorgulescu, A., ‘Pseudo-BL-algebras, I, II’, Mult.-Valued Logic (to appear).Google Scholar
[7]Dvurečenskij, A., ‘Pseudo MV-algebras are intervals in ℓ-groups’, J. Austral. Math. Soc. 72 (2002), 427–445.CrossRefGoogle Scholar
[8]Dvurečenskij, A. and Pulmannová, S., New trends in quantum structures (Kluwer, Dordrecht, 2000).CrossRefGoogle Scholar
[9]Dvurečenskij, A. and Vetterlein, T., ‘Pseudoeffect algebras. I. Basic properties’, Internat. J. Theor. Phys. 40 (2001), 685–701.CrossRefGoogle Scholar
[10]Dvurečenskij, A. and Vetterlein, T., ‘Pseudoeffect algebras. II. Group representations’, Internat. J. Theor. Phys. 40 (2001), 703–726.CrossRefGoogle Scholar
[11]Foulis, D. J. and Bennett, M. K., ‘Effect algebras and unsharp quantum logics’, Found. Phys. 24 (1994), 1325–1346.CrossRefGoogle Scholar
[12]Georgescu, G. and lorgulescu, A., ‘Pseudo-MV algebras: A non-commutative extension of MV- algebras’, in: Proc. Fourth Inter Symp. on Econ. Inform., May 6–9, 1999, Bucharest (ed. Smeureanu, I. et al. ) (INFOREC Printing House, Bucharest, 1999) pp. 961–968.Google Scholar
[13]Georgescu, G. and lorgulescu, A., ‘Pseudo-MV algebras’, Mult-Valued Logic 6 (2001), 95–135.Google Scholar
[14]Goodearl, K. R., Partially ordered abelian groups with interpolation, Math. Surveys and Monographs 20 (Amer. Math. Soc., Providence, RI, 1986).Google Scholar
[15]Greechie, R. J., Foulis, D. J. and Pulmannová, S., ‘The center of an effect algebra’, Order 12 (1995), 91–106.CrossRefGoogle Scholar
[16]Jakubík, J., ‘Cantor-Bernstein theorem for MV-algebras’, Czechoslovak Math. J. 49 (1999), 517–529.CrossRefGoogle Scholar
[17]Jakubík, J., ‘A theorem of Cantor-Bernstein theorem type for orthogonally σ-complete pseudo MValgebras’, Tatra Mt. Math. Publ. (to appear).Google Scholar
[18]Jenča, G., ‘A Cantor-Bernstein type theorem for effect algebras’, Algebra Universalis (to appear).Google Scholar
[19]Kôpka, F. and Chovanec, F., ‘D-posets’, Math. Slovaca 44 (1994), 21–34.Google Scholar
[20]Mundici, D., ‘Interpretation of AF C*-algebras in Łukasiewicz sentential calculus’, J. Funct. Anal. 65 (1986), 15–63.CrossRefGoogle Scholar
[21]Rachůnek, J., ‘A non-commutative generalization of MV-algebras’, Czechoslovak Math. J. 52 (2002), 255–273.CrossRefGoogle Scholar