Riesz space-valued states on pseudo MV-algebras

Anatolij Dvurečenskij

paper

Riesz space-valued states on pseudo MV-algebras

arXiv:1709.06502

Abstract

We introduce Riesz space-valued states, called $(R, 1_{R})$ -states, on a pseudo MV-algebra, where $R$ is a Riesz space with a fixed strong unit $1_{R}$ . Pseudo MV-algebras are a non-commutative generalization of MV-algebras. Such a Riesz space-valued state is a generalization of usual states on MV-algebras. Any $(R, 1_{R})$ -state is an additive mapping preserving a partial addition in pseudo MV-algebras. Besides we introduce $(R, 1_{R})$ -state-morphisms and extremal $(R, 1_{R})$ -states, and we study relations between them. We study metrical completion of unital $ℓ$ -groups with respect to an $(R, 1_{R})$ -state. If the unital Riesz space is Dedekind complete, we study when the space of $(R, 1_{R})$ -states is a Choquet simplex or even a Bauer simplex.