Jun Zhang and Yitong Sun
Abstract
Intransitive indifference relation is prevalent in social and behavioral sciences. One of
its manifestation is the "threshold" phenomena characteristic of
comparative judgments in psychophysics. A formal axiomatic approach to
intransitive indifference was pioneered by Luce (1956), who introduced the
concept of "semiorder". It was later known that any set endowed
with a semiorder relation carries a numerical representation of
fixedintervals (ScottSuppes representation) justifying the commonly adopted
"just noticeable difference" representation of threshold in sensory
psychophysics. Moreover, generalizing semiorder is the socalled interval
order (Fishburn) with a variableinterval representation for its elements;
the latter closely related to biorder (Falmagne, Doignon, etc) that underlies
Guttman scale. I will review the literature, and discuss how an interval
order induces a “nesting” relation, the absence of which characterizes
semiorder. I will then show how pairwise separation of elements of an
intervalordered set (including semiordered set as a special case) can be
linked to topological separation of a suitably constructed topology on such
posets (partiallyordered sets).
