发表机构
Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice（科希策帕沃尔·约瑟夫·沙法里克大学理学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为上下文感知模糊系统中的条件聚合算子族建立纤维表示理论，刻画全局可表示性条件，并统一Choquet、Sugeno与Shilkret积分方案，应用于证据融合与多准则决策。
AI 中文摘要
条件聚合算子根据可容许的上下文来评估配置文件，其中活动源、标准或规则可能变化。对于固定上下文，此类算子实际上是受限配置上的普通单调泛函。非平凡结构出现在族层面，其中不同配置文件空间上的局部泛函必须进行比较，并可能由一个全局规则表示。我们为条件聚合算子族发展了一种纤维表示理论。在分离纤维性质与跨上下文兼容性之后，我们刻画了全局可表示性，而不假设整个全域本身是可容许上下文。当且仅当其局部值在规范零扩展后尊重逐点序时，这样的族由单调全局泛函生成；当此成立时，显式的最小和最大全局生成元可用。这产生了一个层级结构，其包含关系在任意族、零扩展一致族和全局可表示族之间可能严格。对于有限系统，我们随后建立了部分容度的极值扩展定理，并推导了将局部容度粘合为全局容度的精确重叠准则。最后，校准的保表示积分方案统一了Choquet、Sugeno和Shilkret情形：零扩展一致性等价于局部集函数的射影兼容性，而重叠兼容的局部容度产生尖锐的上下模糊积分分数。证据融合和多准则应用将这些结果解释为在非活动源下的鲁棒性、上下文相关交互以及不完整跨上下文信息下的结构不确定性。
英文摘要
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.
Comments23 pages
