Topological aspects in formal concept analysis

Elisabeth Restrepo-Parra, Juan Carlos-Riano-Rojas and Johana Ramirez Gaviria

Topological aspects in formal concept analysis

Elisabeth Restrepo-Parra, Juan Carlos-Riano-Rojas and Johana Ramirez Gaviria

National University of Colombia, Colombia

: J Comput Eng Inf Technol

Abstract

This work presents a theoretical development from a topological approach to formal concept analysis (FCA). This seeks to combine the FCA and a topological study, which enables find information in tables underlying binary and fuzzy data, and hidden information without the use of topological tools. We proposed a method for the analysis of data more accurately in contrast to using only FCA. The basic theorem on concept lattices ensures that formal concepts have complete lattice structure, is discussed alongside the main results of FCA as an area of applied mathematics on databases. The topological structure for formal contexts is proposed from topological basis for the set of objects and attributes. With this in mind to determine relationships between objects and attributes, some topological operators such as interior, closure and boundary for the data were characterized. The continuity between formal contexts with its topological structure was also studied and it describes the representation of formal context as a bipartite graph and the topologies of its associated lattice. We presented a generalization for fuzzy formal concept analysis (FFCA) showing that the results of the classical FCA are preserved and it has extended the topological structure from binary case to fuzzy case. Finally, we applied our methodology in examples of the state of the art. The conclusions were presented, including the fact that knowing the formal concepts of a context, you can quickly extract the topological bases proposed to provide topological structure for the table, also concluded that the generalization for fuzzy data is possible, but has great limited by the lack of specialized software to perform the necessary computations. As possible future work, we propose to develop algorithms for computations in fuzzy large volumes of data, using other topologies and exploring more relationships between the FCA, lattice, graph and topology theory.