2017/08/21 by Dvurečenskij, Anatolij, Zahiri, Omid
#06C15 #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1708.06091
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 ℓ-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.