Relation-based Galois connections: towards the residual of a relation

Inma P. Cabrera, Pablo Cordero, Manuel Ojeda-Aciego, Relation-based Galois connections: towards the residual of a relation, CMMSE 2017: Proceedings of the 17th International Conference on Mathematical Methods in Science and Engineering ( ISBN: 978-84-617-8694-7) , pp. 469--475We explore a suitable generalization of the notion of Galois connection in which their components are binary relations. Many different approaches are possible depending both on the (pre-)order relation between subsets in the underlying powerdomain and the chosen type of relational composition.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

On the notion of fuzzy adjunctions between fuzzy orders

Las adjunciones (también denominadas conexiones de Galois isótonas) entre dos estructuras matemáticas proporcionan una manera de conectar ambas teorías que permite compartir las ventajas de ambas. Hay varios resultados en la literatura previa acerca de las condiciones necesarias y suficientes para la existencia de conexiones de Galois entre dos conjuntos parcialmente ordenados. En otros trabajos anteriores, los autores han estudiado la existencia y construcción del adjunto por la derecha de una aplicación dada entre conjuntos preordenados o dotados de un orden difuso, entendido éste como una relación binaria difusa satisfaciendo reflexividad, transitividad y antisimetría. Se entiende el término adjunción difusa como una pareja de aplicaciones entre dos conjuntos crisp que están dotados de órdenes difusos para las culaes se verifica la condición ρ(a,g(b)) = ρ(f(a),b). Esta definición no está suficientemente justificada en un contexto difuso puesto que las aplicaciones que se consideran se dan entre conjuntos clásicos. En este trabajo se explica la forma en la que la citada definición está relacionada con las funciones difusas, mostrando así que la definición sí es adecuada en ambiente difuso también. La extensión natural de la noción difusa de adjunción contempla dos posibilidades, según si uno considera igualdades difusas o equivalencias difusas asociadas al orden difuso o no.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Extending FuzAtAnalyzer to approach the management of classical negation

FuzAtAnalyzer was conceived as a Java framework which goes beyond of classical tools in formal concept analysis. Specifically, it successfully incorporated the management of uncertainty by means of methods and tools from the area of fuzzy formal concept analysis. One limitation of formal concept analysis is that they only consider the presence of properties in the objects (positive attributes) as much in fuzzy as in crisp case. In this paper, a first step in the incorporation of negations is presented. Our aim is the treatment of the absence of properties (negative attributes). Specifically, we extend the framework by including specific tools for mining knowledge combining crisp positive and negative attributes.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Removing redundancy for attribute implications in data with grades

Reasoning with if-then rules –in particular, with those taking from of implications between conjunctions of attributes– is crucial in many disciplines ranging from theoretical computer science to applications. One of the most important problems regarding the rules is to remove redundancies in order to obtain equivalent implicational sets with lower size.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

On the existence of right adjoints for surjective mappings between fuzzy structures0

En este trabajo los autores continúan su estudio de la caracterización de la existencia de adjunciones (conexiones de Galois isótonas) cuyo codominio no está dotado de estructura en principio. En este artículo se considera el caso difuso en el que se tiene un orden difuso R definido en un conjunto A y una aplicación sobreyectiva f:A-> B compatible respecto de dos relaciones de similaridad definidas en el dominio A y en el condominio B, respectivamente. Concretamente, el problema es encontrar un orden difuso S en B y una aplicación g:B-> A compatible también con las correspondientes similaridades definidas en A y en B, de tal forma que el par (f,g) constituya un adjunción

From an implicational system to its corresponding D-basis

Closure system is a fundamental concept appearing in several areas such as databases, formal concept analysis, artificial intelligence, etc. It is well-known that there exists a connection between a closure operator on a set and the lattice of its closed sets. Furthermore, the closure system can be replaced by a set of implications but this set has usually a lot of redundancy inducing non desired properties. In the literature, there is a common interest in the search of the mini- mality of a set of implications because of the importance of bases. The well-known Duquenne-Guigues basis satisfies this minimality condition. However, several authors emphasize the relevance of the optimality in order to reduce the size of implications in the basis. In addition to this, some bases have been defined to improve the computation of closures relying on the directness property. The efficiency of computation with the direct basis is achieved due to the fact that the closure is computed in one traversal. In this work, we focus on the D-basis, which is ordered-direct. An open problem is to obtain it from an arbitrary implicational system, so it is our aim in this paper. We introduce a method to compute the D-basis by means of minimal generators calculated using the Simplification Logic for implications.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Supported by Grants TIN2011-28084 and TIN2014-59471-P of the Science and Innovation Ministry of Spain, which is co-financed by the European Social Fund

Fuzzy algebras of concepts

Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts.Partial funding for open access charge: Universidad de Málag

CAISL: Simplification Logic for Conditional Attribute Implications

In this work, we present a sound and complete axiomatic system for conditional attribute implications (CAIs) in Triadic Concept Analysis (TCA). Our approach is strongly based on the Simplification paradigm which offers a more suitable way for automated reasoning than the one based on Armstrong’s Axioms. We also present an automated method to prove the derivability of a CAI from a set of CAI s.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Fuzzy Adjunctions revisited

En este trabajo se intenta obtener la noción de adjunción más débil entre estructuras difusas. Este trabajo continúa la línea de investigación en el estudio y construcción d adjunciones que han realizado los autores en contribuciones anteriores. Nos centraremos ahora en la noción de relación difusa que es en cierto sentido interpretable como una función difusa. Existen varios trabajos en la literatura relacionados con este tema. Entre todos ellos, trabajaremos con un enfoque próximo al de Ciric et al cuando definen las denominadas funciones parciales difusas. El nuevo concepto estudiado es el de relaciones difusas funcionales y la construcción de adjunciones entre ellas.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Simplification logic for the management of unknown information

This paper aims to contribute to the extension of classical Formal Concept Analysis (FCA), allowing the management of unknown information. In a preliminary paper, we define a new kind of attribute implications to represent the knowledge from the information currently available. The whole FCA framework has to be appropriately extended to manage unknown information. This paper introduces a new logic for reasoning with this kind of implications, which belongs to the family of logics with an underlying Simplification paradigm. Specifically, we introduce a new algebra, named weak dual Heyting Algebra, that allows us to extend the Simplification logic for these new implications. To provide a solid framework, we also prove its soundness and completeness and show the advantages of the Simplification paradigm. Finally, to allow further use of this extension of FCA in applications, an algorithm for automated reasoning, which is directly built from logic, is defined.Funding for open access charge: Universidad de Málaga / CBUA This article is Supported by Grants TIN2017-89023-P, PRE2018-085199 and PID2021-127870OB-I00 of the Ministry of Science and Innovation of Spain and UMA2018-FEDERJA-001 of the Junta de Andalucia and European Social Fund