partial fuzzy relational equations implicative model, borderline casesalpha-liftsolvability degreeundefined valuespartial fuzzy set, theory partial algebras

Systems of partial fuzzy relational equations employing undefined values in the antecedents and consequents have been approached recently. The primary focus was on the issues of sufficient solvability and solvability criteria. This study introduces another perspective, investigating the behavior of solvability degrees of these systems. We employ operations from the Lower estimation and Dragonfly partial algebras developed in the partial fuzzy set theory framework. Initially, we establish a degree of solvability in an appropriate space of approximations containing potential solutions for the systems. Subsequently, we introduce the concept of the alpha-lift for a given partial fuzzy set and provide its fundamental properties. This concept is employed to modify the antecedents and consequents of a given system of partial fuzzy relational equations, resulting in a modified system. The solvability degree of this modified system is then studied, and we demonstrate that, under sufficient conditions, it significantly enhances the solvability degree of the initial system. This positive impact is observed in the G\"{o}del algebra, the underlying algebraic structure of partial algebras. In conclusion, we provide illustrative examples that effectively demonstrate the theoretical results.