Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T10:28:52.854Z Has data issue: false hasContentIssue false
  • English
  • Français

Pseudo MV-algebras are intervals in ℓ-groups

Part of: General logic Algebraic logic

Published online by Cambridge University Press:  09 April 2009

Anatolij Dvurečenskij
Anatolij Dvurečenskij
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Šefánikova 49, SK-814 73 Bratislava, Slovakia e-mail: [email protected]
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.

We show that any pseudo MV-algebra is isomorphic with an interval Γ(G, u), where G is an ℓ-group not necessarily Abelian with a strong unit u. In addition, we prove that the category of unital ℓ-groups is categorically equivalent with the category of pseudo MV-algebras. Since pseudo MV-algebras are a non-commutative generalization of MV-algebras, our assertions generalize a famous result of Mundici for a representation of MV-algebras by Abelian unital ℓ-groups. Our methods are completely different from those of Mundici. In addition, we show that any Archimedean pseudo MV-algebra is an MV-algebra.

Keywords

pseudo MV-algebraMV-algebraunital ℓ-groupstrong unitcategorical equivalenceArchimedean pseudo MV-algebras

MSC classification

Secondary: 03B50: Many-valued logic 03G12: Quantum logic
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 427 - 446
Copyright
Copyright © Australian Mathematical Society 2002

References

[1] Baer, R., ‘Free sums of groups and their generalizations. An analysis of the associative law’, Amer J. Math. 41 (1949), 706–742.CrossRefGoogle Scholar
[2] Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Publ. 25 (Amer. Math. Soc., Providence, RI, 1997).Google Scholar
[3] Bosbach, B., ‘Concerning semiclans’, Arch. Math. 37 (1981), 316–324.Google Scholar
[4] Chang, C. C., ‘Algebraic analysis of many-valued logics’, Trans. Amer Math. Soc. 88 (1958), 467–490.CrossRefGoogle Scholar
[5] Cignoli, R., D'Ottaviano, I. M. L. and Mundici, D., Algebraic foundations of many-valued reasoning (Kluwer, Dordrecht, 2000).Google Scholar
[6] Darnel, M. R., Theory of lattice-ordered groups (Marcel Dekker, New York, 1995).Google Scholar
[7] Dvurečenskij, A., ‘On partial addition in pseudo 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. 952–960.Google Scholar
[8] Dvurečenskij, A. and Graziano, M. G., ‘Remarks on representations of minimal clans’, Tatra Mt. Math. Publ. 15 (1998), 31–53.Google Scholar
[9] Dvurečenskij, A. and Pulmannová, S., New trends in quantum structures (Kluwer, Dordrecht and Ister Science, Bratislava, 2000).CrossRefGoogle Scholar
[10] Fuchs, L., Partially ordered algebraic systems (Pergamon Press, Oxford, 1963).Google Scholar
[11] Georgescu, G. and Iorgulescu, A., ‘Pseudo MV algebras’, Mult.-Valued Log. 6 (2001), 95–135.Google Scholar
[12] Höhle, U., ‘Commutative, residuated 1-monoids’, in: Non-classical logic and their representation to fuzzy subsets (eds. Höhle, U. and Klement, E. P.) (Kluwer, Dordrecht, 1995) pp. 53–106.CrossRefGoogle Scholar
[13] MacLane, S., Categories for the working mathematician (Springer, New York, 1971).Google Scholar
[14] Mundici, D., ‘Interpretation of AF C*-algebras in Lukasiewicz sentential calculus’, J. Funct. Anal. 65 (1986), 15–63.Google Scholar
[15] Rachůnek, J., ‘A non-commutative generalization of MV-algebras’, Czechoslovak Math. J., to appear.Google Scholar
[16] Schmidt, J., ‘Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Menge’, Arch. Math. 7 (1956), 241–249.Google Scholar
[17] Wyler, O., ‘Clans’, Compos. Math. 17 (1966/1967), 172–189.Google Scholar