In the mathematics
field of order theory
, a containment order
is the partial order
that arises as the subset
-containment relation on some collection of objects. In a simple way, every poset P
= ( X
,≤) is (isomorphism
to) a containment order (just as every group is isomorphic to a permutation group - see Cayley's theorem). To see this, associate to each element x
then the transitivity of ≤ ensures that for all a and b in X, we have
There can be sets of cardinal number less than such that P is isomorphism to the containment order on S. The size of the smallest possible S is called the 2-dimension of S.
Several important classes of poset arise as containment orders for some natural collections, like the Boolean lattice Qn, which is the collection of all 2 n 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.