On closure operators related to maximal tricliques in tripartite hypergraphs

Triadic Formal Concept Analysis (3FCA) was introduced by Lehman and Wille almost two decades ago. Many researchers work in Data Mining and Formal Concept Analysis using the notions of closed sets, Galois and closure operators, and closure systems. However, a proper closure operator for enumeration of triconcepts, i.e. maximal triadic cliques of tripartite hypergraphs, was not introduced. In this paper, we show that the previously introduced operators for obtaining triconcepts and maximal connected and complete sets (MCCSs) are not always consistent and provide the reader with a definition of valid closure operator and associated set system. Moreover, we study the difficulties of related problems from order-theoretic and combinatorial point view as well as provide the reader with justifications of the complexity classes of these problems.