Open Access
Pointwise aggregation of maps: its structural functional equation and some applications to social choice theory
(Elsevier, 2017) Miguel Turullols, Laura de; Campión Arrastia, María Jesús; Candeal, Juan Carlos; Induráin Eraso, Esteban; Paternain Dallo, Daniel; Automática y Computación; Matemáticas; Automatika eta Konputazioa; Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
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-2fuzzy 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.
Open Access
Assigning numerical scores to linguistic expressions
(MDPI, 2017) Campión Arrastia, María Jesús; Falcó Díaz de Cerio, Edurne; García Lapresta, José; Induráin Eraso, Esteban; Institute for Advanced Research in Business and Economics - INARBE; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through distances in a graph plus some suitable penalization of imprecision. The relationship between this setting and the classical problems of numerical representability of orderings, as well as extension of orderings from a set to a superset is also explored. Finally, aggregation procedures of qualitative rankings and scorings are also analyzed.
Open Access
Multidistances and inequality measures on abstract sets: an axiomatic approach
(Elsevier, 2021) Campión Arrastia, María Jesús; Díaz, Irene; Induráin Eraso, Esteban; Martín, Javier; Mayor, Gaspar; Montes Rodríguez, Susana; Raventós Pujol, Armajac; Estatistika, Informatika eta Matematika; Institute for Advanced Research in Business and Economics - INARBE; Estadística, Informática y Matemáticas
Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section.
Embargo
Binary relations coming from solutions of functional equations: orderings and fuzzy subsets
(World Scientific Publishing Company, 2017) Campión Arrastia, María Jesús; Miguel Turullols, Laura de; García Catalán, Olga Raquel; Induráin Eraso, Esteban; Abrísqueta Usaola, Francisco Javier; Automatika eta Konputazioa; Matematika; Institute of Smart Cities - ISC; Institute for Advanced Research in Business and Economics - INARBE; Institute for Advanced Materials and Mathematics - INAMAT2; Automática y Computación; Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We analyze the main properties of binary relations, defined on a nonempty set, that arise in a natural way when dealing with real-valued functions that satisfy certain classical functional equations on two variables. We also consider the converse setting, namely, given binary relations that accomplish some typical properties, we study whether or not they come from solutions of some functional equation. Applications to the numerical representability theory of ordered structures are also furnished as a by-product. Further interpretations of this approach as well as possible generalizations to the fuzzy setting are also commented. In particular, we discuss how the values taken for bivariate functions that are bounded solutions of some classical functional equations define, in a natural way, fuzzy binary relations on a set.
Open Access
Weightable quasi-metrics related to fuzzy sets
(Hacettepe University (Turquía), 2018) Campión Arrastia, María Jesús; García Catalán, Olga Raquel; Induráin Eraso, Esteban; Valero, Óscar; Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Institute for Advanced Research in Business and Economics - INARBE; Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We show that the definition of a fuzzy set is directly related to the existence of a weightable quasi-metric on a universe. This relationship is also explored in terms of functional equations coming either from the membership function of a fuzzy set or from the disymmetry function of a quasi-metric.
Open Access
An axiomatic approach to finite means
(Elsevier, 2018) Campión Arrastia, María Jesús; Candeal, Juan Carlos; García Catalán, Olga Raquel; Giarlotta, Alfio; Induráin Eraso, Esteban; Estatistika, Informatika eta Matematika; Institute for Advanced Research in Business and Economics - INARBE; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications.
Open Access
Strong negations and restricted equivalence functions revisited: an analytical and topological approach
(Elsevier, 2021) Bustince Sola, Humberto; Campión Arrastia, María Jesús; Miguel Turullols, Laura de; Induráin Eraso, Esteban; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Throughout this paper, our main idea is to analyze the concepts of a strong negation and a restricted equivalence function, that appear in a natural way when dealing with theory and applications of fuzzy sets and fuzzy logic. Here we will use an analytical and topological approach, showing how to construct them in an easy way. In particular, we will also analyze some classical functional equation related to those key concepts.
Open Access
Comparison meaningful operators and ordinal invariant preferences
(Elsevier, 2015) Candeal, Juan Carlos; Induráin Eraso, Esteban; Matemáticas; Matematika
The existence of a continuous and order-preserving real-valued function, for the class of continuous and ordinal invariant total preorders, defined on the Banach space of all bounded real-valued functions, which are in turn defined on a given set Ω, is characterized. Whenever the total preorder is nontrivial, the type of representation obtained leads to a functional equation that is closely related to the concept of comparison meaningfulness, and is studied in detail in this setting. In particular, when restricted to the space of bounded and measurable real-valued functions, with respect to some algebra of subsets of Ω, we prove that, if the total preorder is also weakly Paretian, then it can be represented as a Choquet integral with respect to a {0,1}-valued capacity. Some interdisciplinary applications to measurement theory and social choice are also considered.
Open Access
Decomposition of fuzzy relations: an application to the definition, construction and analysis of fuzzy preferences
(MDPI, 2023) Campión Arrastia, María Jesús; Induráin Eraso, Esteban; Raventós Pujol, Armajac; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Institute for Advanced Research in Business and Economics - INARBE
In this article, we go deeper into the study of some types of decompositions defined by triangular norms and conorms. We work in the spirit of the classical Arrovian models in the fuzzy setting and their possible extensions. This allows us to achieve characterizations of existence and uniqueness for such decompositions. We provide rules to obtain them under some specific conditions. We conclude by applying the results achieved to the study of fuzzy preferences.
Open Access
Past-myopic economic agents
(Economics Bulletin, 2025-03-30) Campión Arrastia, María Jesús; Induráin Eraso, Esteban; Munárriz Iriarte, Ana; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Research in Business and Economics - INARBE; Institute for Advanced Materials and Mathematics - INAMAT2
In this paper, the idea of myopic preference and myopic topology provided by Brown and Lewis is further explored. In this sense, new definitions of myopia are introduced for finite spaces. The main contribution of the work is the inclusion of the past in the models. We have defined the notions of past-myopic preference and topology for sequence spaces and n-dimensional spaces. Adding this new dimension makes it possible to work with decision spaces where the economic agent only has information about past events and when she has to choose accordingly to it, which is in line with the reality of certain economic situations, such as voting or finances. This approach generates a wide room for future research lines related to the idea of myopia. Based on this, it would allow to study in forthcoming research past-hyperopic topologies and preferences, and to interconnect different preferences, defined on the past and future models.