States on EMV-algebras
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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.
KeywordsMV-algebra EMV-algebra State State-morphism Krein–Mil’man representation Pre-state Strong pre-state Jordan signed measure Integral representation of states The Horn–Tarski theorem
Author A.D. has received research Grants from the Slovak Research and Development Agency under contract APVV-16-0073 and the Grant VEGA No. 2/0069/16 SAV.
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All authors have declared that they have no conflict of interest.
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This article does not contain any studies with human participants or animals performed by any of the authors.
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