Partial quasi-metrics and fixed point theory: an aggregation viewpoint

Abstract

In many fields of applied sciences, the aggregation of numerical values, in order to get a final one which allows to make a decision, plays a central role. Many times these numerical values represent dissimilarities and the merged value can be interpreted as a global dissimilarity. Inspired, on the one hand, by the interest that causes the dissimilarities aggregation problem and, on the other hand, by the utility of generalized dissimilarities in applied sciences, we focus our work on the problem of merging the so-called partial quasi-metrics, which have been introduced in the literature with the aim of developing a framework that allows to unify the notion of metric, quasi-metric and partial metric under a unique one. Concretely, we characterize those functions that merge a collection of partial quasi-metrics into a new one. Moreover, a few relationships between this kind of functions and those that merge (quasi-)metrics and partial metrics are discussed. Furthermore, a general fixed point result for contractions obtained through aggregation functions is given.

Keywords:

• Partial quasi-metric
• aggregation
• contraction
• fixed point

Acknowledgments

The authors acknowledge financial support from FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación/${}_{-}$Proyecto PGC2018-095709-B-C21, and by Spanish Ministry of Economy and Competitiveness under contract DPI2017-86372-C3-3-R (AEI, FEDER, UE). This work is also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direcció General d'Innovació i Recerca, Govern de les Illes Balears) and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union's Horizon 2020 research and innovation programme under grant agreements No 779776 and No 871260, respectively. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by European Commission [grant number 779776], [grant number 871260] and Direcció General d'Innovació i Recerca, Govern de les Illes Balears [grant number PROCOE/4/2017] and Spanish Ministry of Economy and Competitiveness [grant number DPI2017-86372-C3-3-R] and Ministerio de Ciencia, Innovación y Universidades Agencia Estatal de Investigación [grant number PGC2018-095709-B-C21].

Notes on contributors

Pilar Fuster-Parra

Pilar Fuster-Parra received a BS degree and a Ms Degree in Mathematics by the University of Valencia in 1988. She received her PhD in Computer Sciences by the University of Balearic Islands in 1996. She was a visiting researcher at Laboratoire d'Analyse et d'Architecture des systèmes (LAAS) of CNRS in Toulouse, France. She is currently an Associate Professor at the University of Balearic Islands (Department of Mathematics and Computer Sciences). She is a member of the research group MOTIBO of University of Balearic Islands. Her research interests lie primarily in the areas of mathematical modeling, Artificial Intelligence, fuzzy sets, possibility theory, generalized distances, multi-agent systems.

Juan José Miñana

Juan-José Miñana is graduated in Mathematics by the University of Valencia from 2010. Then, in 2015 he obtained a PhD in Mathematics by the Polytechnical University of Valencia. Since September of 2018 is Associate Professor (tenure track) in the Department of Mathematics and Computer Science at Balearic Islands University. His research interest includes different topics related to both Pure and Applied Mathematics. In particular, his research has focused in Fuzzy Mathematics and its applications to Computer Science, Artificial Intelligence, Engineering and Economics. He has been the author of more than 30 research papers and has contributed as a reviewer in different scientific journals. He is a member of the research group MOTIBO of the Balearic Islands University.

Óscar Valero

Oscar Valero Sierra received a BS degree and a Ms degree in Mathematics by the University of Valencia in 2000. Later on, in 2003, he received a Ph.D. in Mathematics by the Polytechnical University of Valencia. He was a visiting researcher at “Center for Efficiency-Oriented Languages” which belongs to the Computer Science Department at National University of Ireland. He is currently an Associate Professor in the Department of Mathematics and Computer Science at Balearic Islands University. So far, he has taught lectures on bachelor's degrees in Engineering, Business Administration, Economics and Mathematics. His research activity focus on Mathematical Models applied to Computer Science, Artificial Intelligence, Engineering and Economics. In this direction, he has published more than 140 papers and he serves as a member of editorial board of 9 journals. He is the Principal Investigator of the research group MOTIBO of the Balearic Islands University. A detailed information about his research activity can be viewed at http://www.uib.eu/research/groups/grup/MOTIBO/.