## The $$\sigma$$-complete MV-algebras which have enough states.(English)Zbl 1081.06011

The authors characterize Łukasiewicz tribes, special sets of fuzzy sets forming $$\sigma$$-complete MV-algebras closed under pointwise limits of monotone functions. To have this characterization, they generalize the concept of McNaughton functions to $$\sigma$$-McNaughton functions as the closure of McNaughton functions under countable MV-algebraic operations. In addition, the authors characterize $$\sigma$$-complete MV-algebras to be isomorphic to Łukasiewicz tribes as those having a separating system of pure states.

### MSC:

 06D35 MV-algebras 03E72 Theory of fuzzy sets, etc. 28E10 Fuzzy measure theory
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