On the Homologies of Multi-ary Relations

We construct a complex \(K^n\) of \(m\)-ary relations, \(1 \leq m \leq n+1\), in a finite set \(X \neq 0\), representing a model of an abstract cellular complex. For such a complex \(K^n\) we define the matrices of incidence and coincidence, the groups of homologies \(\mathcal{H}_m(K^n)\) and cohomologies \(\mathcal{H}^m(K^n)\) on the group of integers \(\mathbf{Z}\), and the Euler characteristic. On a combinatorial basis we derive their main properties. In further publications we will derive more analogues of classical properties, and also applications with respect to the existence of fixed relations in the utilization of the isomorphisms will be investigated. In particular, we intend to complete the theory of hypergraphs with the help of such topological observations.