On uniquely partitionable relational structures and object systems

We introduce object systems as a common generalization of graphs, hypergraphs, digraphs and relational structures. Let C be a concrete category, a simple object system over C is an ordered pair S = (V,E), where E = {A₁,A₂,...,Aₘ} is a finite set of the objects of C, such that the ground-set ${\displaystyle V\left(A_i\right)}$ of each object ${\displaystyle A_i\in E}$ is a finite set with at least two elements and ${\displaystyle V\supseteq {\bigcup }_{i=1}^{m}V\left(A_i\right)}$. To generalize the results on graph colourings to simple object systems we define, analogously as for graphs, that an additive induced-hereditary property of simple object systems over a category C is any class of systems closed under isomorphism, induced-subsystems and disjoint union of systems, respectively. We present a survey of recent results and conditions for object systems to be uniquely partitionable into subsystems of given properties.