# Containment order

In the mathematical 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 (isomorphic 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* of *X* the set

then the transitivity of ≤ ensures that for all *a* and *b* in *X*, we have

There can be sets of cardinality less than such that *P* is isomorphic 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 *Q ^{n}*, 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**.

## See also

## References

- Fishburn, P.C.; Trotter, W.T. (1998). "Geometric containment orders: a survey".
*Order*.**15**(2): 167–182. doi:10.1023/A:1006110326269. - Santoro, N., Sidney, J.B., Sidney, S.J., and Urrutia, J. (1989). "Geometric containment and partial orders".
*SIAM Journal on Discrete Mathematics*.**2**(2): 245–254. doi:10.1137/0402021.