Submeasures on nuanced MV-algebras

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 NMVA n is defined and studied and the notion of a bounded local submeasure on a perfect NMVA n is introduced. It is proved that any bounded local submeasure on a perfect NMVA n L can be extended to a submeasure on L.