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Containment order

In the field of , a containment order is the that arises as the -containment relation on some collection of objects. In a simple way, every poset P = ( X,≤) is ( 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

X_\leq(x) = \{ y \in X | y \leq x\} ;

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

X_\leq(a) \subseteq X_\leq(b) \mbox{ precisely when } a \leq b .

There can be sets S of less than |X| such that P is 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 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 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

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