States on EMV-algebras

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. We show that states always exist, 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 $\mathrm{\ell }$-group. Finally, we show that every state can be represented by a unique regular probability measure, and a variant of the Horn–Tarski theorem is proved.