Triadic fuzzy Galois connections as ordinary connections

We present results on triadic Galois connections and trilattices associated with ternary fuzzy relations. These structures are fundamental in the analysis of three-way relational data. We provide an axiomatization of triadic fuzzy Galois connections and establish two ways of representing these structures by their ordinary counterparts—one via a Cartesian representation and the other via a-cuts. The results allow us to easily transfer some of the known results from the ordinary case to the fuzzy case. This transfer is illustrated by an alternative proof of the basic theorem of fuzzy concept trilattices via reduction to the ordinary basic theorem. In addition, we provide a hint to generalize the presented results to n-ary structures which are of increasing importance in data analysis of n-way data.