二元关系的对偶合成及其在传递性中的应用

The Dual Composition of Binary Relations and Its Applications to Transitivity Properties

系统地讨论了偏好结构理论中的各种传递性质,引入了二元关系的一种新的合成运算:对偶合成.结果表明,这种对偶合成可以方便地刻画反向传递性,它与合成运算一起可以刻画半传递性和Ferrers传递性.利用二元关系的合成和对偶合成运算建立了二元关系的各种类型的传递性质的若干等价条件.这些等价条件都是用集合的包含式表示的,这种表示有利于判断一个二元关系是否具有某种传递性质.

This paper is a comprehensive discussion of several transitivity properties in the theory of preference modelling. The concept of the dual composition of binary relations is introduced. The dual composition can be used to characterize negative transitivity, and it can be used with the composition to characterize semi-transitivity and Ferrets property. Several equivalent conditions of some types of transitivity properties are established via the composition and the dual composition. All the equivalent conditions are in the form of set inclusions which enable us to easily judge if a binary relation has certain transitivity property.