Interval-valued Atanassov intuitionistic t-norms and t-conorms endowed with the usual or admissible orders

The interval-valued fuzzy sets and Atanassov intuitionistic fuzzy sets can be extended to a more general framework to simultaneously deal with uncertainty in both membership and non-membership values. This fact leads to the concept of interval-valued Atanassov intuitionistic fuzzy sets (IVAIFS), as given by Atanassov and Gargov (Fuzzy Sets Syst, 31(3):343–349, 1989). In this paper, we focus on the study of interval-valued Atanassov intuitionistic t-norms and t-conorms, studying important properties and characterizations of some of their sub-classes. In addition, we do not only consider just the usual order on IVAIFS, but also admissible orders. Finally, we establish the basis for the use of this study in the approximate reasoning context.