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Loomis-sikorski theorem for σ-complete MV-algebras and 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 Štefánikova 49 SK-814 73 Bratislava Slovakia e-mail: dvurecen@mat.savba.sk
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Abstract

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We show that every σ-complete MV-algebra is an MV-σ-homomorphic image of some σ-complete MV- algebra of fuzzy sets, called a tribe, which is a system of fuzzy sets of a crisp set Ω containing 1Ω and closed under fuzzy complementation and formation of min {∑nfn, 1}. Since a tribe is a direct generalization of a σ-algebra of crisp subsets, the representation theorem is an analogue of the Loomis-Sikorski theorem for MV-algebras. In addition, this result will be extended also for Dedekind σ-complete ℓ-groups with strong unit.

Keywords

maximal idealstateσ-complete MV-algebrabascially disconnected spacetribeLoomis-Sikorski theoremℓ-groupg-tribe.

MSC classification

Secondary: 03B50: Many-valued logic 03G12: Quantum logic
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 2 , April 2000 , pp. 261 - 277
Copyright
Copyright © Australian Mathematical Society 2000

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