Resumo

MACEDA-MENDEZ, Adolfo. Fuzzy mathematics and cubical complexes. Rev. Mat [online]. 2017, vol.24, n.2, pp.201-225. ISSN 1409-2433.  http://dx.doi.org/10.15517/rmta.v24i2.29872.

Fuzzy mathematics generalize concepts of traditional mathematics using fuzzy sets. This enables to study and model more properly phenomens characterized by imprecision. These generalizations includes concepts of algebra, analysis and topology. On the other side, cubical complexes have applications in digital image processing and in the study of dynamical systems, but in the actual literature there is not an extension of their properties using fuzzy sets. In this paper is proposed a generalization of the concept of cubical complex and of some of their properties, such as connectedness, polyhedral realization, connected component and holes, using fuzzy sets. The upper and lower trees of a fuzzy cubical complex are defined, which give information about the way in which its regional extrema are related. The homology groups of a fuzzy cubical complex are defined and it is shown that the rank of the 0-homology group of a given level is equal with the number of regional maxima of that level. Finally, it is shown how to associate a fuzzy cubical complex with a bidimensional digital grayscale image in order to study somo of its topological properties.

Palavras-chave : cubical complex; fuzzy topology; connectedness.

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