A holistic approach to the composition of ternary relations

In this paper, we present a systematic approach to the study of the composition of ternary relations from the point of view of the degrees of freedom available when linking a 3-tuple to two given 3-tuples. We propose a way of enumerating all possible 4-point compositions (one degree of freedom) and 5-point compositions (two degrees of freedom) of ternary relations, and establish a correspondence between them. Furthermore, we identify the associative compositions and explore interesting mixed-associativity cases. Finally, we use the tools of projection and cylindrical extension to relate the 4-point and 5-point compositions of ternary relations to the 3-point compositions of binary relations.