Inclusion order

${\displaystyle X_{\leq ($x)}=\{y\in X\mid y\leq x\};}

then the transitivity of ≤ ensures that for all a and binX, we have

${\displaystyle X_{\leq ($a)}\subseteq X_{\leq (b)}{\text{ precisely when }}a\leq b.}

There can be sets ${\displaystyle S}$ofcardinality less than ${\displaystyle |X|}$ such that Pisisomorphic to the inclusion order on S. The size of the smallest possible S is called the 2-dimensionofP.

Several important classes of poset arise as inclusion 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.

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