有界分配格上的拟多项式函数
Quasi-polynomial functions over bounded distributive lattices
摘要：
在[arXiv 0811.3913]中，作者引入了拟多项式函数的概念，作为在有界链X上定义和赋值的映射f：X^n -> X，并且可以分解为f（x_1，.，x_n）=p（phi（x_1），.，φ（x_n）），其中p是多项式函数（即，使用链运算和的变量和常数的组合是保序映射。在目前的文件中，我们研究这一概念在更一般的设置的基础域和codomain集，可能是不同的，有界分配格，并在内部功能不一定是为了保持。这些功能自然出现在不确定性下的决策范围内，因为，如本文所示，他们subsequently与Sugeno积分，其变量由一个给定的效用函数变换的整体偏好泛函。为了公理化拟多项式函数类，我们提出了聚集理论中已建立的性质的几个推广，并表明[arXiv 0811.3913]中给出的一些特征在这种一般设置中仍然成立。此外，我们研究了所谓的转换多项式函数（实质上是一元映射与多项式函数的合成），并表明，在一定条件下，它们减少到拟多项式函数。
英文摘要：
In [arXiv 0811.3913] the authors introduced the notion of quasi-polynomial function as being a mapping f: X^n -> X defined and valued on a bounded chain X and which can be factorized as f(x_1,...,x_n)=p(phi(x_1),...,phi(x_n)), where p is a polynomial function (i.e., a combination of variables and constants using the chain operations / and) and phi is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [arXiv 0811.3913] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions.
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