## 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.

### MSC:

 06D35 MV-algebras 28C99 Set functions and measures on spaces with additional structure 46A55 Convex sets in topological linear spaces; Choquet theory

### Keywords:

state; semisimple MV-algebra; clan; Bauer simplex
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### References:

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