Submeasures on nuanced MV-algebras. (English) Zbl 1315.06013
Summary: This paper further investigates the properties of submeasures on \(n\)-nuanced MV-algebras introduced by the author in a previous work. We prove that there is a one-to-one correspondence between the set of submeasures on an \(n\)-nuanced MV-algebra and the set of submeasures on its MV-center. As a main result, we prove an extension theorem for submeasures on \(n\)-nuanced MV-algebras. This result generalizes the extension theorems which have been proved by Riečan for MV-algebras and by Georgescu for Łukasiewicz-Moisil algebras. The perfect \(\mathrm{NMVA}_n\) is defined and studied and the notion of a bounded local submeasure on a perfect \(\mathrm{NMVA}_n\) is introduced. It is proved that any bounded local submeasure on a perfect \(\mathrm{NMVA}_n\) \(L\) can be extended to a submeasure on \(L\).
MSC:
| 06D35 | MV-algebras |
| 03G25 | Other algebras related to logic |
| 28B15 | Set functions, measures and integrals with values in ordered spaces |
Keywords:
nuanced MV-algebras; exhaustive property; monotone sets; local submeasures; extension theoremReferences:
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