More on decompositions of a fuzzy set in fuzzy topological spaces (Q2815611)

Autors' abstract: Using new properties of the concept of fuzzy points in the sense of Pu Pao-Ming and Liu Ying-Ming, we first prove that every fuzzy set \(\lambda\not=0\) is decomposed by two fuzzy sets \(\lambda_{\mathcal{O}(X,\sigma^f)}\) and \(\lambda^\ast_{\mathcal{PC}(X,\sigma^f )}\), where \((X,\sigma^f)\) is a specified Chang's fuzzy space. Namely, \(\lambda = \lambda_{\mathcal{O}(X,\sigma^f )}\vee \lambda^\ast_{\mathcal{PC}(X,\sigma^f )}\) and \(\lambda_{\mathcal{O}(X,\sigma^f )}\wedge \lambda^\ast_{\mathcal{PC}(X,\sigma^f )}=0\) hold, and the fuzzy set \(\lambda_{\mathcal{O}(X,\sigma^f)}\) is fuzzy open in \((X,\sigma^f)\). Finally, these results are applied to the case where \(X =\mathbb{Z}^n (n > 0)\) and \(\sigma^f = (\kappa^n)^f\), where the topological space \((X, \sigma)\) is the digital \(n\)-space \((\mathbb{Z}^n, \kappa^n)\).