Many interesting examples of posets have ground sets consisting of a family of geometric objects. A circle order is a partially ordered set P with the property that to each element x1 ∈ P we can assign a circle disk c1 (i.e. a circle together with its interior) so that x1 x2 if and only if c 1 ⊂ c2. A spherical order is a poset of k-1 dimensional spheres in Rk ordered by containment. It turns out that there are many different kinds of posets that are actually circle and spherical orders. Because of this fact, research based solely on circle and spherical orders has been active and fruitful. For some time it was conjectured that every finite three-dimensional poset was a spherical order. It wasn't until 1999, Felsner, Fishburn and Trotter proved this conjecture is false. However, the question as to when is a finite three-dimensional poset a circle or spherical order remains open.; In my research a partial answer is obtain for this question by introducing a new method for representing a finite poset by a graph. This new graph is called a Jordan graph and we proved that there is a one-to-one correspondence between the Jordan graphs and disjoint circle orders. Also, we define a category of disjoint circle orders, list three relations among the morphisms and proved that these relations are the complete set of relations for the morphisms. We illustrate that this category can be useful in a different setting by showing that this category is also a category of the Loop Braid group. We extend the definition of Jordan graph to the generalized Jordan graph. The generalized Jordan graph yields a structure on a finite poset that allow us to draw some conclusions as to when the poset is a circle or spherical order.
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