Conditional Aggregation in Context-Aware Fuzzy Systems: A Fibered Representation Theory
Conditional aggregation operators evaluate profiles relative to admissible contexts whose active sources, criteria, or rules may vary. For a fixed context, such an operator is indeed an ordinary monotone functional on the restricted profile. The nontrivial structure appears at the family level, where local functionals on different profile spaces must be compared and possibly represented by one global rule. We develop a fibered representation theory for families of conditional aggregation operators. After separating fiberwise properties from cross-context compatibility, we characterize global representability without assuming that the full universe is itself an admissible context. Such a family is generated by a monotone global functional if and only if its local values respect the pointwise order after canonical zero extension; when this holds, explicit least and greatest global generators are available. This yields a hierarchy whose inclusions can be strict between arbitrary, zero-extension-consistent, and globally representable families. For finite systems, we then establish an extremal extension theorem for partial capacities and derive an exact overlap criterion for gluing local capacities into global ones. Finally, calibrated representation-preserving integral schemes unify the Choquet, Sugeno, and Shilkret cases: zero-extension consistency is equivalent to projective compatibility of local set functions, while overlap-compatible local capacities yield sharp lower and upper fuzzy-integral scores. Evidence-fusion and multi-criteria applications interpret these results as robustness under inactive sources, context-dependent interaction, and structural uncertainty under incomplete cross-context information.