Kybernetika 38 no. 3, 297-318, 2002

States on unital partially-ordered groups

Anatolij Dvurečenskij


We study states on unital po-groups which are not necessarily commutative as normalized positive real-valued group homomorphisms. We show that in contrast to the commutative case, there are examples of unital po-groups having no state. We introduce the state interpolation property holding in any Abelian unital po-group, and we show that it holds in any normal-valued unital $\ell$-group. We present a connection among states and ideals of po-groups, and we describe extremal states on the state space of unital po-groups.