In the mathematical field of order theory, an inclusion order is the partial order that arises as the subset-inclusion relation on some collection of objects. In a simple way, every poset P = (X,≤) is (isomorphic to) an inclusion order (just as every group is isomorphic to a permutation group – see Cayley's theorem). To see this, associate to each element x of X the set
X\leq(x)=\{y\inX\midy\leqx\};
then the transitivity of ≤ ensures that for all a and b in X, we have
X\leq(a)\subseteqX\leq(b)preciselywhena\leqb.
There can be sets
S
|X|
Several important classes of poset arise as inclusion orders for some natural collections, like the Boolean lattice Qn, which is the collection of all 2n subsets of an n-element set, the interval-containment orders, which are precisely the orders of order dimension at most two, and the dimension-n orders, which are the containment orders on collections of n-boxes anchored at the origin. Other containment orders that are interesting in their own right include the circle orders, which arise from disks in the plane, and the angle orders.