Uncertainty measures, decomposability and admissibility
Fuzzy Sets and Systems - Special memorial volume on mathematical aspects of fuzzy set theory
Conditioning on MV-algebras and additive measures—I
Fuzzy Sets and Systems - Special issue on fuzzy measures and integrals
Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning
Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning
Uncertainty measures---Problems concerning additivity
Fuzzy Sets and Systems
Uncertainty measures---Problems concerning additivity
Fuzzy Sets and Systems
Measure-free conditioning and extensions of additive measures on finite MV-algebras
Fuzzy Sets and Systems
Extensions of states on MV-quantales
Fuzzy Sets and Systems
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We consider extensions of the unique additive measure on a finite MV-chain to uncertainty measures on its canonical Girard algebra extension. If the underlying MV-chain has more than two non-trivial elements, in a previous paper we have proved the non-existence of strongly additive measure extensions, where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so-called divisible disjoint unions. This negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all MV-subalgebras of the canonical Girard algebra extension. We obtain a characterization of all such MV-subalgebras which are in fact MV-chains and, using this result, we come up to a characterization of all weakly additive measure extensions. Moreover, we characterize also all the valuation extensions and we prove the incompatibility of these two types of measure extensions, except for underlying MV-chains with one or two or four non-trivial elements.