Type 〈1,1〉〈1,1〉 fuzzy quantifiers determined by fuzzy measures defined on residuated lattices. Part II. Permutation and isomorphism invariances

We study the properties of permutation invariance and isomorphism invariance of fuzzy quantifiers of type 〈1,1〉〈1,1〉 defined using fuzzy measures and integrals. These properties hold if fuzzy quantifiers are invariant with respect to permutations (bijective mappings) on the universe of discourse (permutation invariance) and with respect to bijections between different universes of discourse (isomorphism invariance). In other words, fuzzy quantifiers with these properties are sensitive only to the cardinality of subsets of the universe of discourse, and not to the individual nature of the elements of these subsets. We characterize these properties by means of the corresponding properties of functionals used in the definition of the fuzzy quantifiers.