Componentwise decomposition of some lattice-valued fuzzy integrals
For each complete lattice \(L\) and any measurable space \((X,{\mathcal A})\) one can introduce the Sugeno integral of a measurable function \(f:X \to L\) with respect to a monotone set function \(m:{\mathcal A}\to L\), Sug\(( m,f)=\sup\{\inf\{a,m(\{f\geq a\})\mid a\in L\}\). The paper deals with a complete lattice \(L\) equipped with an involutive negation \(n\) (complement) allowing to introduce a new lattice \({\mathcal L}=\{(a,b)\in L^2\mid a\leq n(b) \}\). Observe that \(\{(a,n(a))\mid a\in L\}\) is a complete sublattice of \({\mathcal L}\) isomorphic to \(L\). The main result of the paper shows that the Sugeno integral in \({\mathcal L}\) can be represented by means of two Sugeno integrals in \(L\). Several other properties of the Sugeno integral in \({\mathcal L}\) are discussed, especially in the specific case when \(L=([0,1],\vee, \wedge)\). An application to synthetic evaluation of objects is included.
scientific article; zbMATH DE number 5128487
scientific article