States on EMV-algebras

Abstract

We define a state as a [0, 1]-valued, finitely additive function attaining the value 1 on an EMV-algebra, which is an algebraic structure close to MV-algebras, where the top element is not assumed. The state space of an EMV-algebra is a convex space that is not necessarily compact, and in such a case, the Krein–Mil’man theorem cannot be used. Nevertheless, we show that the set of extremal states generates the state space. We show that states always exist and the extremal states are exactly state-morphisms. Nevertheless, the state space is a convex space that is not necessarily compact; a variant of the Krein–Mil’man theorem, saying states are generated by extremal states, is proved. We define a weaker form of states, pre-states and strong pre-states, and also Jordan signed measures which form a Dedekind complete $$\ell $$-group. Finally, we show that every state can be represented by a unique regular Borel probability measure, and a variant of the Horn–Tarski theorem is proved.