Sufficient Solvability Conditions for Systems of Partial Fuzzy Relational Equations

This paper focuses on searching sufficient conditions for the solvability of systems of partial fuzzy relational equations. In the case of solvable systems, we provide solutions of the systems. Two standard systems of fuzzy relational equations – namely the systems built on the basic composition and on the Bandler-Kohout subproduct – are considered under the assumption of partiality. Such an extension requires to employ partial algebras of operations for dealing with undefined values. In this investigation, we consider seven most-known algebras of undefined values in partial fuzzy set theory such as the Bochvar, Bochvar external, Sobociński, McCarthy, Nelson, Kleene, and the Łukasiewicz algebra. Conditions that are sufficient for the solvability of the systems are provided. The crucial role will be played by the so-called boundary condition.