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
Several important classes of poset arise as containment 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.
- Fishburn, P.C. and 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.
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