Summary: A family F of triplets of a set X is a cyclic order if the following axioms are satisfied: \[ \begin{alignedat}{2} (a,b,c)\in F &\Rightarrow (b,c,a) \text{ and }(c,a,b)\in F &&\text{(cyclicity);}\\ (a,b,c)\in F &\Rightarrow (b,a,c)\not\in F &&\text{(antisymmetry);}\\ (a,b,c) \text{ and }(c,d,a)\in F &\Rightarrow (b,c,d) \text{ and }(d,a,b)\in F &&\text{(transitivity).} \end{alignedat} \] Such a concept aims to formalize some problems related to points or intervals drawn on a circle or on the plane, or with geometric representations of finite groups (\((a,b,c)\in F\) is to be read as ‘\(b\) is located between \(a\) and \(c\)’).
We move into the context of the so defined ternary relation some classical questions arising in the theory of partial ordered subsets, and deal, for instance, with extendability problem and with the problem of the minimal partition of a cyclic order into complete cyclic suborders. We also present several applications to such questions as the characterization of cyclic graphs or circular-arc graphs.