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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]
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Abstract

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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

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