Every state on semisimple MV-algebra is integral. (English) Zbl 1107.06007
An MV-algebra \(M\) is semisimple iff it is isomorphic to a clan of fuzzy sets, i.e., a system \({\mathcal S}\) which contains \(1\), and is closed with respect to negation and truncate sum. Moreover, the system of fuzzy sets can be chosen as a system of continuous functions on a Hausdorff compact space. Such an MV-algebra admits a separating system of states. The main result of the paper is the statement saying that every finitely additive state on a semisimple MV-algebra can be represented as an integral with respect to a Borel measure on this compact Hausdorff space.
Reviewer: Anatolij Dvurečenskij (Bratislava)
MSC:
| 06D35 | MV-algebras |
| 28C99 | Set functions and measures on spaces with additional structure |
| 46A55 | Convex sets in topological linear spaces; Choquet theory |
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