States on EMV-algebras
Logic
2017-09-19 v1 Commutative Algebra
Abstract
We define a state as a -valued, finitely additive function attaining the value 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 -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.
Cite
@article{arxiv.1708.06091,
title = {States on EMV-algebras},
author = {Anatolij Dvurečenskij and Omid Zahiri},
journal= {arXiv preprint arXiv:1708.06091},
year = {2017}
}