12 research outputs found
EXISTENCE OF MAXIMAL ELEMENTS OF SEMICONTINUOUS PREORDERS
We discuss the existence of an upper semicontinuous multi-utility representation of a preorder on a topological space. We then prove that every weakly upper semicontinuous preorder is extended by an upper semicontinuous preorder and use this fact in order to show that every weakly upper semicontinuous preorder on a compact topological space admits a maximal element
Szpilrajn-type theorems in economics
The Szpilrajn "constructive type" theorem on
extending binary relations,
or its generalizations by Dushnik and Miller [10],
is one of the best known theorems in
social sciences and mathematical economics.
Arrow [1], Fishburn [11],
Suzumura [22], Donaldson and Weymark [8] and
others
utilize Szpilrajn's
Theorem and the Well-ordering principle to obtain more general "existence type" theorems
on
extending binary relations. Nevertheless, we are generally interested not only
in the existence of linear extensions of a binary relation R, but in something more:
the conditions of the preference sets and the properties which satisfies
to be "inherited" when one passes to any member of some
\textquotedblleft interesting\textquotedblright
family of linear extensions of R.
Moreover,
in extending a preference relation , the problem will often be how to incorporate some additional preference data with a minimum
of disruption of the existing structure or how to extend the relation so that some desirable new condition is fulfilled. The key to addressing these kinds of problems is
the szpilrajn constructive method.
In this
paper, we give two general
"constructive type" theorems on
extending binary relations, a
Szpilrajn type and a Dushnik-Miller
type theorem, which generalize and give a "constructive type" version of all the well known extension
theorems in the literature
Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness
Looking at decisiveness as crucial, we discuss the existence of an order-preserving function
for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation.
We further present conditions for the existence of an upper semicontinuous order-preserving function
for a fuzzy binary relation on a crisp topological space
The Axiomatic Structure of Empirical Content
In this paper, we provide a formal framework for studying the empirical content of a given theory. We define the falsifiable closure of a theory to be the least weakening of the theory that makes only falsifiable claims. The falsifiable closure is our notion of empirical content. We prove that the empirical content of a theory can be exactly captured by a certain kind of axiomatization, one that uses axioms which are universal negations of conjunctions of atomic formulas. The falsifiable closure operator has the structure of a topological closure, which has implications, for example, for the behavior of joint vis a vis single hypotheses.
The ideas here are useful for understanding theories whose empirical content is well-understood (for example, we apply our framework to revealed preference theory, and Afriat's theorem), but they can also be applied to theories with no known axiomatization. We present an application to the theory of multiple selves, with a fixed finite set of selves and where selves are aggregated according to a neutral rule satisfying independence of irrelevant alternatives. We show that multiple selves theories are fully falsifiable, in the sense that they are equivalent to their empirical content
Normally preordered spaces and utilities
In applications it is useful to know whether a topological preordered space
is normally preordered. It is proved that every -space equipped with
a closed preorder is a normally preordered space. Furthermore, it is proved
that second countable regularly preordered spaces are perfectly normally
preordered and admit a countable utility representation.Comment: 17 pages, 1 figure. v2 contains a second proof to the main theorem
with respect to the published version. The last section of v1 is not present
in v2. It will be included in a different wor