Date: Friday, February 22, 2013
Location: 1084 East Hall (3:00 PM to 4:00 PM)
Title: Topological characterization of interval and semi-orders
Abstract: The concept of semi-order was introduced by Luce (1956) to capture the intransitive indifference relation prevalent in social and behavioral sciences. Its numerical representation manifests a threshold structure (Scott-Suppes representation) characteristic of comparative judgments in psychophysics. Later, it became known that semi-order (and its fixed-threshold representation) was a special case of the more general interval order (and its interval graph representation) as succinctly characterized by Fishburn. In this talk, we first show how interval order induces a "nesting" relation, a partial order itself. A set with a semi-order on it is then precisely an interval-ordered set that does not contain any nesting among its elements. When nesting occurs, an interval-ordered set has two lexicographic orders, which agree on the subset of elements that do not nest one-another. Next, we investigate topologies on interval-order sets, and construct a topology (based on the notion of upper- and lower-holdings) that allows us to relate topological axiomatic separations to order relations. Specifically, under our proposed topology, two distinct elements are (i) nested iff they are T0 but not T1 separated; (ii) indifferent but non-nested iff they are T1 but not T2 separated; (iii) comparable iff they are T2 separated. Therefore, we achieve topological characterization of pairwise relations of all points in an interval-order set in terms of their topological separability.
(Work done with student collaborator Yitong Sun)
Speaker: Jun Zhang
Institution: Psychology, University of Michigan
Event Organizer: Peter Miller millerpd@umich.edu
|
