Complete solution sets of inf-→ interval-valued fuzzy relation equations

Fuzzy relation equations play an important role in fuzzy set theory. Interval-valued fuzzy set theory is an extension of fuzzy theory in which a closed subinterval of the unit interval is assigned with membership degree. Therefore, it is very significant to study interval-valued fuzzy relation equations from both the theoretical and practical viewpoints. In this paper, the solution sets of interval-valued fuzzy relation equations with inf-→ composition is investigated, where → is interval-valued R-, S- or QL-implication. Necessary and sufficient conditions such that there exist solutions for these equations are first shown. Some sufficient conditions for existence of maximal solutions for these equations are represented, and then it is shown that the complete solution sets of inf-→ interval-valued fuzzy relation equations can be determined by their maximal solutions. Finally, the solution sets of linear interval-valued fuzzy relation equations are described by a method similar to that in linear algebra.