Non-riumerical fuzzy and possibilistic measures taking their values in
partially ordered sets, semilattices or lattices are introduced. Using the operations of supremurn and infimum in these structures, the inner and outer (lower and upper) extensions of the original measures are investigated and defined. The conditions under which the resulting functions -extend conservatively the original ones and possess the properties of fuzzy or possibilistic measures, are explicitly stated and relevant assertions are proved.
When proposing and processing uncertainty decision-making algorithms of various kinds and purposes, we more and more often meet probability distributions ascribing non-numerical uncertainty degrees to random events. The reason is that we have to process systems of uncertainties for which the classical conditions like σ-additivity or linear ordering of values are too restrictive to define sufficiently closely the nature of uncertainty we would like to specify and process. In cases of non-numerical uncertainty degrees, at least the following two criteria may be considered. The first criterion should be systems with rather complicated, but sophisticated and nontrivially formally analyzable uncertainty degrees, e. g., uncertainties supported by some algebras or partially ordered structures. Contrarily, we may consider easier relations, which are non-numerical but interpretable on the intuitive level. Well-known examples of such structures are set-valued possibilistic measures. Some specific interesting results in this direction are introduced and analyzed in this contribution.