Pseudo MV-algebras are intervals in $\ell$-groups

J. Austral. Math. Soc. 72 (2002), 427-445

Pseudo MV-algebras are intervals in $\ell$ -groups

Anatolij Dvurecenskij
Mathematical Institute
Slovak Academy of Sciences
Stefánikova 49
SK-814 73 Bratislava
Slovakia
dvurecen@mat.savba.sk

Abstract

We show that any pseudo MV-algebra is isomorphic with an interval $\Gamma(G,u)$ , where

is an $\ell$ -group not necessarily Abelian with a strong unit

. In addition, we prove that the category of unital $\ell$ -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 $\ell$ -groups. Our methods are completely different from those of Mundici. In addition, we show that any Archimedean pseudo MV-algebra is an MV-algebra.

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