Triadic concept lattices of data with graded attributes

We present foundations for triadic concept analysis of data with graded attributes. Triadic concept analysis departs from the dyadic case by taking into account modi, such as time instances or conditions, under which objects have attributes. That is, instead of a two-dimensional table filled with 0s and 1s (equivalently, binary relation or two-dimensional binary matrix), the input data in triadic concept analysis consists of a three-dimensional table (equivalently, ternary relation or three-dimensional binary matrix). In the ordinary triadic concept analysis, one assumes that the ternary relationship between objects, attributes, and modi, which specifies whether a given object has a given attribute under a given modus, is a yes-or-no relationship. In the present paper, we show how triadic concept analysis may be developed in a setting in which the ternary relationship between objects, attributes, and modi is a matter of degree rather than a yes-or-no relationship. We present the basic notions for the new, “graded” setting, an illustrative example, and generalize basic results including the basic theorem of triadic concept analysis.