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Loomis-Sikorski theorem for monotone σ-complete effect algebras

  • Anatolij Dvurečenskij (a1)

Abstract

We show that monotone σ -complete effect algebras under some conditions are σ - homomorphic images of effect-tribes (as monotone σ -complete effect algebras), which are nonempty systems of fuzzy sets closed under complements, sums of fuzzy sets less than 1, and containing all pointwise limits of nondecreasing fuzzy sets. Because effect-tribes are generalizations of Boolean σ -algebras of subsets, we present a generlization of the Loomis-Sikorski theorem for such effect algebras. We show that we can choose an effect-tribe to be a system of affin fuzzy sets. In addition, we present a new version of the Loomis-Sikorski theorem for σ-complete MV-algebras.

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References

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[1]Chang, C. C., ‘Algebraic analysis of many valued logics’, Trans. Amer. Math. Soc. 88 (1958), 467490.
[2]Dvurečenskij, A., ‘Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups’, preprint.
[3]Dvurečenskij, A., ‘Loomis-Sikorski theorem for σ-complete MV-algebras and ℓ-groups’, J. Austral. Math. Soc. Ser. A 68 (2000), 261277.
[4]Dvuterčenskij, A and Pulmannová, S., New trends in quantum structures (Kluwer Acad. Publ., Dordrecht, Ister Science, Bratislava, 2000).
[5]Dvurečenskij, A. and Vetterlein, T., ‘Pseudoeffect algebras. II. Group representation’, Int. J. Theor. Phys. 40 (2001), 703726.
[6]Foulis, D. J. and Bennett, M. K., ‘Effect algebras and unsharp quantum logics’, Found. Phys. 24 (1994), 13251346.
[7]Goodearl, K. R., Partially ordered Abelian groups with interpolation, Math. Surveys Monogr. 20 (Amer. Math. Soc., Providence, RI, 1986).
[8]Kôpka, F. and Chovanec, F., ‘D-posets’, Math. Slovaca 44 (1994), 2134.
[9]Kuratowaski, K., Topology l, (in Russian) (Mir, Moskva, 1966).
[10]Mundici, D., ‘Interpretation of AF C*-algebras in Łukasiewicz sentential calculus’, J. Funct. Anal. 65 (1986), 1563.
[11]Mundici, D., ‘Tensor products and the Loomis-Sikorski theorem for MV-algebras’, Adv. Appl. Math. 22 (1999), 227248.
[12]Ravindran, K., On a structure theory of effect algebras (Ph. D. Thesis, Kansas State Univ., Manhattan, Kansas, 1996).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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