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Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects yi conform in some consistent way to those existing among the corresponding represented objects xi. Somewhat more formally, for a set of properties and relations, a -representation of some structure X is a structure Y that is the image of X under a homomorphism that preserves . The label representation is sometimes also applied to the homomorphism itself.
Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representation theory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces.
Many partial orders arise from (and thus can be represented by) collections of geometric objects. Among them are the n-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-called circle orders, the posets representable in terms of containment among disks in the plane. A particularly nice result in this field is the characterization of the planar graphs as those graphs whose vertex-edge incidence relations are circle orders.