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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 = is an inclusion order. To see this, associate to each element x of X the set
then the transitivity of ≤ ensures that for all a and b in X, we have
There can be sets S {\displaystyle S} of cardinality less than | X | {\displaystyle |X|} such that P is isomorphic to the inclusion order on S. The size of the smallest possible S is called the 2-dimension of P.
Several important classes of poset arise as inclusion orders for some natural collections, like the Boolean lattice Q, which is the collection of all 2 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.