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ON THE LOOMIS–SIKORSKI THEOREM FOR MV-ALGEBRAS WITH INTERNAL STATE

Part of: Boolean algebras (Boolean rings) Distributive lattices Algebraic logic

Published online by Cambridge University Press:  01 April 2011

A. DI NOLA ,
A. DVUREČENSKIJ  and
A. LETTIERI
A. DI NOLA
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, I-84084 Fisciano, Salerno, Italy (email: [email protected])
A. DVUREČENSKIJ*
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia (email: [email protected])
A. LETTIERI
Affiliation:
Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli Federico II, via Monteoliveto 3, I-80134 Napoli, Italy (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In Flaminio and Montagna [‘An algebraic approach to states on MV-algebras’, in: Fuzzy Logic 2, Proc. 5th EUSFLAT Conference, Ostrava, 11–14 September 2007 (ed. V. Novák) (Universitas Ostraviensis, Ostrava, 2007), Vol. II, pp. 201–206; ‘MV-algebras with internal states and probabilistic fuzzy logic’, Internat. J. Approx. Reason.50 (2009), 138–152], the authors introduced MV-algebras with an internal state, called state MV-algebras. (The letters MV stand for multi-valued.) In Di Nola and Dvurečenskij [‘State-morphism MV-algebras’, Ann. Pure Appl. Logic161 (2009), 161–173], a stronger version of state MV-algebras, called state-morphism MV-algebras, was defined. In this paper, we present the Loomis–Sikorski theorem for σ-complete MV-algebras with a σ-complete state-morphism-operator, showing that every such MV-algebra is aσ-homomorphic image of a tribe of functions with an internal state induced by a function where all the MV-operations are defined by points.

Keywords

MV-algebraBoolean algebrastate MV-algebrainternal statestate-morphism MV-algebrastate-morphism-operatorLoomis–Sikorski theoremtribe

MSC classification

Secondary: 06D35: MV-algebras 03G12: Quantum logic
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 3 , December 2010 , pp. 317 - 333
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This research was supported by the Center of Excellence SAS—Quantum Technologies, ERDF OP R & D Projects CE QUTE ITMS 26240120009 and meta-QUTE ITMS 26240120022, the grant VEGA No. 2/0032/09 SAV, by the Slovak Research and Development Agency under the contract APVV-0071-06, Bratislava, and by the Slovak-Italian project SK-IT 0016-08.

References

[1]Barbieri, G. and Weber, H., ‘Measures on clans and on MV-algebras’, in: Handbook of Measure Theory, Vol. II (ed. Pap, E.) (Elsevier Science, Amsterdam, 2002), pp. 911–945.Google Scholar
[2]Buhagiar, D., Chetcuti, E. and Dvurečenskij, A., ‘Loomis–Sikorski representation of monotone σ-complete effect algebras’, Fuzzy Sets and Systems 157 (2006), 683–690.CrossRefGoogle Scholar
[3]Chang, C. C., ‘Algebraic analysis of many-valued logics’, Trans. Amer. Math. Soc. 88 (1958), 467–490.CrossRefGoogle Scholar
[4]Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of Many-valued Reasoning (Kluwer Academic Publishers, Dordrecht, 1998).Google Scholar
[5]Di Nola, A. and Dvurečenskij, A., ‘State-morphism MV-algebras’, Ann. Pure Appl. Logic 161 (2009), 161–173.CrossRefGoogle Scholar
[6]Di Nola, A. and Dvurečenskij, A., ‘On some classes of state-morphism MV-algebras’, Math. Slovaca 59 (2009), 517–534.CrossRefGoogle Scholar
[7]Di Nola, A., Dvurečenskij, A. and Lettieri, A., ‘On varieties of MV-algebras with internal states’, Internat. J. Approx. Reason. 51 (2010), 680–694.CrossRefGoogle Scholar
[8]Di Nola, A., Dvurečenskij, A. and Lettieri, A., ‘Erratum state-morphism MV-algebras’, Ann. Pure Appl. Logic 161 (2010), 1605–1607, Ann. Pure Appl. Logic 161 (2009), 161–173.CrossRefGoogle Scholar
[9]Dvurečenskij, A., ‘Loomis–Sikorski theorem for σ-complete MV-algebras and ℓ-groups’, J. Aust. Math. Soc. Ser. A 68 (2000), 261–277.CrossRefGoogle Scholar
[10]Dvurečenskij, A., ‘States on pseudo-effect algebras with general comparability’, Kybernetika (Prague) 40 (2004), 397–420.Google Scholar
[11]Dvurečenskij, A. and Kalmbach, G., ‘States on pseudo MV-algebras and the hull-kernel topology’, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 50 (2002), 131–146.Google Scholar
[12]Dvurečenskij, A. and Pulmannová, S., New Trends in Quantum Structures (Kluwer Academic Publishers, Dordrecht, 2000).CrossRefGoogle Scholar
[13]Dvurečenskij, A. and Rachůnek, J., ‘On Riečan and Bosbach states for bounded non-commutative Rℓ-monoids’, Math. Slovaca 56 (2006), 487–500.Google Scholar
[14]Flaminio, T. and Montagna, F., ‘An algebraic approach to states on MV-algebras’, in: Fuzzy Logic 2, Proc. 5th EUSFLAT Conference, Ostrava, 11–14 September 2007 (ed. V. Novák) (Universitas Ostraviensis, Ostrava, 2007), Vol. II, pp. 201–206.Google Scholar
[15]Flaminio, T. and Montagna, F., ‘MV-algebras with internal states and probabilistic fuzzy logic’, Internat. J. Approx. Reason. 50 (2009), 138–152.CrossRefGoogle Scholar
[16]Georgescu, G., ‘Bosbach states on fuzzy structures’, Soft Computing 8 (2004), 217–230.CrossRefGoogle Scholar
[17]Goodearl, K. R., Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, 20 (American Mathematical Society, Providence, RI, 1986).Google Scholar
[18]Jakubík, J., ‘On complete MV-algebras’, Czechoslovak Math. J. 45(120) (1995), 473–480.CrossRefGoogle Scholar
[19]Kroupa, T., ‘Every state on semisimple MV-algebra is integral’, Fuzzy Sets and Systems 157 (2006), 2771–2782.CrossRefGoogle Scholar
[20]Kühr, J. and Mundici, D., ‘De Finetti theorem and Borel states in [0,1]-valued algebraic logic’, Internat. J. Approx. Reason. 46 (2007), 605–616.CrossRefGoogle Scholar
[21]Kuratowski, K., Topology I (Mir, Moskva, 1966) (in Russian).Google Scholar
[22]Loomis, L. H., ‘On the representation of σ-complete Boolean algebras’, Bull. Amer. Math. Soc. 53 (1947), 757–760.CrossRefGoogle Scholar
[23]Mundici, D., ‘Interpretation of AF C *-algebras in Łukasiewicz sentential calculus’, J. Funct. Anal. 65 (1986), 15–63.CrossRefGoogle Scholar
[24]Mundici, D., ‘Averaging the truth-value in Łukasiewicz logic’, Studia Logica 55 (1995), 113–127.CrossRefGoogle Scholar
[25]Mundici, D., ‘Tensor products and the Loomis–Sikorski theorem for MV-algebras’, Adv. Appl. Math. 22 (1999), 227–248.CrossRefGoogle Scholar
[26]Panti, G., ‘Invariant measures in free MV-algebras’, Comm. Algebra 36 (2008), 2849–2861.CrossRefGoogle Scholar
[27]Riečan, B., ‘On the probability on BL-algebras’, Acta Math. Nitra 4 (2000), 3–13.Google Scholar
[28]Riečan, B. and Mundici, D., ‘Probability on MV-algebras’, in: Handbook of Measure Theory, Vol. II (ed. Pap, E.) (Elsevier Science, Amsterdam, 2002), pp. 869–909.Google Scholar
[29]Sikorski, R., ‘On the representation of Boolean algebras as fields of sets’, Fund. Math. 35 (1948), 247–256.CrossRefGoogle Scholar
[30]Sikorski, R., Boolean Algebras, 2nd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 2, 25 (Springer, Berlin–Göttingen–Heidelberg, 1964).Google Scholar