Triadic concept lattices in the framework of aggregation structures

Triadic concept analysis is a method of analysis of three-way tabular data describing objects by binary attributes they have under various conditions. It aims to extract a set of three-dimensional clusters, so-called triadic concepts, from the data. While the method was recently extended to handle the case of graded data, this extension is only one of a variety of possible generalizations, namely the one where the so-called antitone concept-forming operators are considered. The generalizations not covered by the mentioned approach include isotone concept-forming operators, one sided concepts, the use of truth stressing hedges and many others. Since all of the generalizations have their applications, for example in factor analysis of relational data, it is desirable to have a uniform mathematical foundations covering all of the cases. In this paper we provide such foundations. By using triadic aggregation structures as a scale of truth degrees we develop triadic concept analysis in a very general setting that covers a wide variety of its possible extensions in a uniform way.