Functionally Constructed Semi Linear Copulas from Fuzzy Implication Operators

This paper proposes a framework for generating symmetric two-branched copulas directly from fuzzy implications via monotone function composition. The proposed approach is based on a two-branched composition principle induced by fuzzy implications and yields four explicit construction schemes for broad classes of genuine copulas. Under suitable analytical assumptions, the resulting copulas satisfy the boundary conditions, symmetry, and global 2-increasingness. The framework also establishes a structural relation between fuzzy implications and the corresponding copulas. In contrast to related approaches that often remain at the level of quasi-copulas or semicopulas, the present method produces fully valid copulas while allowing distinct monotone behavior in each branch. The resulting families inherit analytical properties from the underlying fuzzy implications and may exhibit strong dependence, as reflected by high values of Spearman’s ρ and Kendall’s τ.