完全分配格上的不变泛函
Invariant functionals on completely distributive lattices
摘要：
本文主要研究定义在完全分配格上的泛函，这些泛函在保{任意}联和交的映射下是不变的。我们证明了非减不变泛函的类相一致的类Sugeno积分与$\{0,1\}$值的能力，所谓的长期泛函，从而扩展到无限的情况下，以及境界完全分配格以前的结果。此外，我们还证明了，在完全链上的泛函的情况下，非减条件是多余的。Sugeno积分类的特征，以及它的超类包括所有的多项式泛函，提供了通过显示的公理化（在齐性方面），他们的限制有限泛函仍然持有完全分配格。我们还提出了完全分配格上多项式泛函的规范范式表示，它表现为它们的有限对应物的自然扩张，并且作为副产品，我们得到了有界格情况下完全分配性的公理化。
英文摘要：
In this paper we are interested in functionals defined on completely distributive lattices and which are invariant under mappings preserving {arbitrary} joins and meets. We prove that the class of nondecreasing invariant functionals coincides with the class of Sugeno integrals associated with $\{0,1\}$-valued capacities, the so-called term functionals, thus extending previous results both to the infinitary case as well as to the realm of completely distributive lattices. Furthermore, we show that, in the case of functionals over complete chains, the nondecreasing condition is redundant. Characterizations of the class of Sugeno integrals, as well as its superclass comprising all polynomial functionals, are provided by showing that the axiomatizations (given in terms of homogeneity) of their restriction to finitary functionals still hold over completely distributive lattices. We also present canonical normal form representations of polynomial functionals on completely distributive lattices, which appear as the natural extensions to their finitary counterparts, and as a by-product we obtain an axiomatization of complete distributivity in the case of bounded lattices.
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