Fuzzy relational equations; Partial Fuzzy Set Theory; Dragonfly Algebra

The article focuses on a problem that naturally comes to an interest by joining two classical topics, namely the solvability of systems of fuzzy relational equations, and the partiality as a tool for modeling undefined values. As the first topic basically formally investigates the correctness of fuzzy rule-based systems by the investigation of the preservation of modus ponens, the other one allows dealing in an elegant algebraic way with distinct undefined values, in the particular case of the Dragonfly algebra, the focus is on missing values, their connection is straightforward. Indeed, dealing with missing values is rather omnipresent, and distinct expert rule-based systems are not immune to it so, such an investigation is highly desirable. What is not so straightforward are the results ensuring the solvability of such systems, i.e., the existence of safe models of the rules. Some previous results have been published and they relied on restrictions on the algebraic level. This article brings a new insight and investigates, what happens if we refuse to accept such a restriction. The answer is interesting as restricting the choice of the algebra does not seem to be critical but it imposes some restrictions on the sides of the consequents. Luckily, these restrictions are not so critical and restrictive from the application point of view.

The article focuses on a problem that naturally comes to an interest by joining two classical topics, namely the solvability of systems of fuzzy relational equations, and the partiality as a tool for modeling undefined values. As the first topic basically formally investigates the correctness of fuzzy rule-based systems by the investigation of the preservation of modus ponens, the other one allows dealing in an elegant algebraic way with distinct undefined values, in the particular case of the Dragonfly algebra, the focus is on missing values, their connection is straightforward. Indeed, dealing with missing values is rather omnipresent, and distinct expert rule-based systems are not immune to it so, such an investigation is highly desirable. What is not so straightforward are the results ensuring the solvability of such systems, i.e., the existence of safe models of the rules. Some previous results have been published and they relied on restrictions on the algebraic level. This article brings a new insight and investigates, what happens if we refuse to accept such a restriction. The answer is interesting as restricting the choice of the algebra does not seem to be critical but it imposes some restrictions on the sides of the consequents. Luckily, these restrictions are not so critical and restrictive from the application point of view.