María Jesús Campión

INTERVAL-VALUED REPRESENTABILITY OF QUALITATIVE DATA: THE CONTINUOUS CASE

In the framework of the representability of ordinal qualitative data by means of interval-valued correspondences, we study interval orders defined on a nonempty set X. We analyse the continuous case, that corresponds to a set endowed with a topology that furnishes an idea of continuity, so that it becomes natural to ask for the existence of quantifications based on interval-valued mappings from the set of data into the real numbers under preservation of order and topology. In the present paper we solve a continuous representability problem for interval orders. We furnish a characterization of the representability of an interval order through a pair of continuous real-valued functions so that each element in X has associated in a continuous manner a characteristic interval or equivalently a symmetric triangular fuzzy number.

Representability of binary relations through fuzzy numbers

We analyse the representability of different classes of binary relations on a set by means of suitable fuzzy numbers. In particular, we show that symmetric triangular fuzzy numbers can be considered as the best codomain to represent interval orders. We also pay attention to the representability of other classes of acyclic binary relations.

AGGREGATION OF PREFERENCES IN CRISP AND FUZZY SETTINGS: FUNCTIONAL EQUATIONS LEADING TO POSSIBILITY RESULTS

We analyze various models introduced in social choice to aggregate individual preferences. We show that on the basis of most of these models there is a system of functional equations such that, in many cases, the origin of impossibility results in a social choice model is the non-existence of a solution for the corresponding system. Among the functional equations considered, we pay a particular attention to general means and associativity, proving that the existence of an associative bivariate mean is equivalent to the existence of a semilatticial partial order. This key result allows us to explain how the knowledge of associative bivariate means can be used to solve social choice paradoxes. In our analysis we deal both with crisp and fuzzy settings.

The existence of utility functions for weakly continuous preferences on a Banach space

Abstract In this note we prove that the weak topology of a Banach space has the Continuous Representability Property which means that every (weakly) continuous total preorder defined on a Banach space can be represented by a weakly continuous utility function. This shows that weak topologies are perfect to look for continuous utility functions that represent preferences on infinite-dimensional commodity spaces.

Isotonies on ordered cones through the concept of a decreasing scale

Abstract Using techniques based on decreasing scales, necessary and sufficient conditions are presented for the existence of a continuous and homogeneous of degree one real-valued function representing a (not necessarily complete) preorder defined on a cone of a real vector space. Applications to measure theory and expected utility are given as consequences.

Order Embeddings with Irrational Codomain: Debreu Properties of Real Subsets

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.

Continuous order representability properties of topological spaces and algebraic structures

In the present paper, we study the relationship between con- tinuous order-representability and the fulllment of the usual covering properties on topological spaces. We also consider the case of some al- gebraic structures providing an application of our results to the social choice theory context.

Pointwise aggregation of maps: its structural functional equation and some applications to social choice theory

Abstract We study a structural functional equation that is directly related to the pointwise aggregation of a finite number of maps from a given nonempty set into another. First we establish links between pointwise aggregation and invariance properties. Then, paying attention to the particular case of aggregation operators of a finite number of real-valued functions, we characterize several special kinds of aggregation operators as strictly monotone modifications of projections. As a case study, we introduce a first approach of type-2 fuzzy sets via fusion operators. We develop some applications and possible uses related to the analysis of properties of social evaluation functionals in social choice, showing that those functionals can actually be described by using methods that derive from this setting.

Reinterpreting a fuzzy subset by means of a Sincov's functional equation

Throughout the paper it is shown that the classical definition of a fuzzy subset carries additional structures. The concept of a fuzzy subset is regarded from an alternative point of view: namely, the characteristic function of a fuzzy subset may be reinterpreted in terms of a Sincov’s functional equation in two variables. Since the solutions of a Sincov’s functional equation are also closely related to the existence of representable total preorders, an special attention is paid to the relationship between fuzzy subsets and total preorders defined on a universe. Some possible applications of this approach are pointed out in the final section

Topological interpretations of fuzzy subsets. A unified approach for fuzzy thresholding algorithms

We show that the classical definition of a fuzzy subset carries additional structures of a topological nature. We look at the concept of a fuzzy subset and its corresponding @a-cuts from an alternative point of view: namely, a fuzzy subset may be interpreted as a nested topology on a crisp set of reference, called a universe. Several kinds of fuzzy subsets associated to this interpretation are analyzed. Other topologies induced by fuzzy subsets are considered, paying special attention to their relationship with total preorders defined on the universe. Moreover, this theoretical approach allows us to provide a unified framework for most of the fuzzy thresholding algorithms that can be found in the literature.