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 Π of properties and relations, a Π-representation of some structure X is a structure Y that is the image of X under a s homomorphism that preserves Π. 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.
Although the term representation theory is well established in the algebraic sense discussed above, there are many other uses of the term representation throughout mathematics.
An active area of graph theory is the exploration of isomorphisms between graphs and other structures. A key class of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets (or, more precisely, non-disjointness) is a symmetric relation. This gives rise to the study of intersection graphs for innumerable families of sets. One foundational result here, due to Paul Erdős and colleagues, is that every n-vertex graph may be represented in terms of intersection among subsets of a set of size no more than n2/4.
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