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1.
The objective of this paper is to investigate the role of the set of irrational numbers as the codomain of order-preserving
functions defined on topological totally preordered sets. We will show that although the set of irrational numbers does not
satisfy the Debreu property it is still nonetheless true that any lower (respectively, upper) semicontinuous total preorder
representable by a real-valued strictly isotone function (semicontinuous or not) also admits a representation by means of
a lower (respectively, upper) semicontinuous strictly isotone function that takes values in the set of irrational numbers.
These results are obtained by means of a direct construction. Moreover, they can be related to Cantor’s characterization of the real line to obtain much more general results on the semicontinuous
Debreu properties of a wide family of subsets of the real line.
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2.
Gianni Bosi Juan Carlos Candeal Esteban Induráin Esteban Oloriz Margarita Zudaire 《Order》2001,18(2):171-190
In the framework of the analysis of orderings whose associated indifference relation is not necessarily transitive, we study the structure of an interval order and its representability through a pair of real-valued functions. We obtain a list of characterizations of the existence of a representation, showing that the three main techniques that have been used in the literature to achieve numerical representations of interval orders are indeed equivalent. 相似文献
3.
Let G be a locally compact group. We show that L1(G) does notcontain any analytic semigroups which are bounded on the half-disc,by using some facts about the analytic Radon-Nikodm propertyin Banach spaces. This solves a question raised by J. Esterle. 相似文献
4.
We analyze the numerical representability of total preorders defined on semitopological real algebras through continuous order-preserving
real-valued functions that are also additive and multiplicative. The results obtained are used to interpret important concepts
arising in Social Choice theory. 相似文献
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We study several functional equations in two variables that are closely related to the functional equation of associativity. We analyze the hierarchy among these equations, paying a particular attention to selections, semi-lattices and weakenings of associativity. 相似文献
7.
In the present paper, first we study in a systematic way the numerical representation problem for total preorders defined either on groups or on real vector spaces. Then, we consider groups and real vector spaces equipped with a topology, and analyze the fulfillment of the so-called continuous representability property; the latter meaning that every continuous total preorder defined on the given topological space admits a continuous real-valued order-preserving function. We also explore the analogous cases as above for total preorders that are compatible with the given algebraic structure, looking for real-valued, continuous or not, order-preserving functions that, in addition, are algebraic homomorphisms. 相似文献
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9.
This paper introduces a different approach to the study of the existence of numerical representations of totally ordered sets (chains). We pay attention to the properties of non-representable chains showing that, under certain conditions, those chains must have a sort of lexicographic behaviour similar to that of the lexicographic plane. We prove that a countably bounded connected chain (Z, \prec )(Z, \prec ) admits a lexicographic decomposition as a subset of the lexicographic product \Bbb R ×Z\Bbb R \times Z. Then we apply our approach to state both a sufficient and a necessary condition for the lack of utility functions. The concept of planar chain is also introduced. 相似文献
10.
Summary Homothetic function is a term which refers to some extension of the concept of a homogeneous function. We study different hierarchies of generalized homogeneous functions. The main result is a general classification of those functions. 相似文献