Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot
Abstract
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 antisymmetric, 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 {\sigma}algebra to be transitive. These results are framed in the registers of order, topology, algebra and measuretheory; and also beyond mathematics in economics: the exploitation of Villegas' notion of monotonic continuity by ArrowChichilnisky in the context of Savage's theorem in decision theory, and the extension of Diamond's impossibility result in social choice theory by BasuMitra. As such, this chapter has a synthetic and expository motivation, and can be read as a plea for interdisciplinary conversations, connections and collaboration.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2007.01952
 arXiv:
 arXiv:2007.01952
 Bibcode:
 2020arXiv200701952K
 Keywords:

 Economics  Theoretical Economics;
 Mathematics  General Topology
 EPrint:
 Positivity and its Applications, 2021