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

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

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: adinola@unisa.it)
A. DVUREČENSKIJ*
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia (email: dvurecen@mat.savba.sk)
A. LETTIERI
Affiliation:
Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli Federico II, via Monteoliveto 3, I-80134 Napoli, Italy (email: lettieri@unina.it)
*
✉For correspondence; e-mail: dvurecen@mat.savba.sk
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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.

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