Inmathematics, arepresentation is a very general relationship that expresses similarities (or equivalences) between mathematical objects orstructures. Roughly speaking, a collectionY of mathematical objects may be said torepresent another collectionX of objects, provided that the properties and relationships existing among the representing objectsyi conform, in some consistent way, to those existing among the corresponding represented objectsxi. More specifically, given a setΠ of properties andrelations, aΠ-representation of some structureX is a structureY that is the image ofX under ahomomorphism that preservesΠ. The labelrepresentation is sometimes also applied to the homomorphism itself (such asgroup homomorphism ingroup theory).[1][2]
Perhaps the most well-developed example of this general notion is the subfield ofabstract algebra calledrepresentation theory, which studies the representing of elements ofalgebraic structures bylinear transformations ofvector spaces.[2]
Although the termrepresentation theory is well established in the algebraic sense discussed above, there are many other uses of the termrepresentation throughout mathematics.
An active area ofgraph theory is the exploration of isomorphisms betweengraphs and other structures.A key class of such problems stems from the fact that, likeadjacency inundirected graphs,intersection of sets(or, more precisely,non-disjointness) is asymmetric relation.This gives rise to the study ofintersection graphs for innumerable families of sets.[3]One foundational result here, due toPaul Erdős and his colleagues, is that everyn-vertex graph may be represented in terms of intersection amongsubsets of a set of size no more thann2/4.[4]
Representing a graph by such algebraic structures as itsadjacency matrix andLaplacian matrix gives rise to the field ofspectral graph theory.[5]
Dual to the observation above that every graph is an intersection graph is the fact that everypartially ordered set (also known as poset) is isomorphic to a collection of sets ordered by theinclusion (or containment) relation ⊆.Some posets that arise as theinclusion orders for natural classes of objects include theBoolean lattices and theorders of dimensionn.[6]
Many partial orders arise from (and thus can be represented by) collections ofgeometric objects. Among them are then-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-calledcircle orders—the posets representable in terms of containment among disks in the plane. A particularly nice result in this field is the characterization of theplanar graphs, as those graphs whose vertex-edge incidence relations are circle orders.[7]
There are also geometric representations that are not based on inclusion. Indeed, one of the best studied classes among these are theinterval orders,[8] which represent the partial order in terms of what might be calleddisjoint precedence of intervals on thereal line: each elementx of the poset is represented by an interval [x1,x2], such that for anyy andz in the poset,y is belowz if and only ify2 <z1.
Inlogic, the representability ofalgebras asrelational structures is often used to prove the equivalence ofalgebraic andrelational semantics. Examples of this includeStone's representation ofBoolean algebras asfields of sets,[9]Esakia's representation ofHeyting algebras as Heyting algebras of sets,[10] and the study of representablerelation algebras and representablecylindric algebras.[11]
Under certain circumstances, a single functionf :X →Y is at once an isomorphism from several mathematical structures onX. Since each of those structures may be thought of, intuitively, as a meaning of the imageY (one of the things thatY is trying to tell us), this phenomenon is calledpolysemy—aterm borrowed from linguistics. Some examples of polysemy include: