论文标题
数学经济学的二元关系：关于Eilenberg，Villegas和Degroot的连续性，添加性和单调性
Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot
论文作者
论文摘要
本章探讨了在数学经济学的两个重要问题中的积极性和顺序如何发挥作用，在这样做的两个重要问题中，对连续性，可添加性和单调性的假设进行了仔细审查。提供了两组结果：与Eilenberg（1941）的第一批出发，对拓扑的必要条件和足够的条件，在拓扑空间上存在反对称，完整，及时和连续的二元关系；第二，从Degroot（1970）的结果中提到了添加性假设，该结果确保了σ-代数上的完整二元关系是传递的。这些结果在秩序，拓扑，代数和措施理论的记录中构建；以及《经济学中的数学》之外：在萨维奇（Savage）定理的决策理论定理的背景下，维拉加（Villegas）对维勒加斯（Villegas）对单调连续性的概念，以及钻石对钻石不可能的扩展导致了Basu-Mitra的社会选择理论。因此，本章具有综合和说明性动机，可以将其视为跨学科对话，联系和协作的恳求。
This chapter examines how positivity and order play out in two important questions in mathematical economics, and in so doing, subjects the postulates of continuity, additivity and monotonicity to closer scrutiny. Two sets of results are offered: the first departs from Eilenberg's (1941) necessary and sufficient conditions on the topology under which an anti-symmetric, complete, transitive and continuous binary relation exists on a topologically connected space; and the second, from DeGroot's (1970) result concerning an additivity postulate that ensures a complete binary relation on a σ-algebra to be transitive. These results are framed in the registers of order, topology, algebra and measure-theory; and also beyond mathematics in economics: the exploitation of Villegas' notion of monotonic continuity by Arrow-Chichilnisky in the context of Savage's theorem in decision theory, and the extension of Diamond's impossibility result in social choice theory by Basu-Mitra. As such, this chapter has a synthetic and expository motivation, and can be read as a plea for inter-disciplinary conversations, connections and collaboration.
