Representation (mathematics)
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. 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''. More specifically, given 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
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
that preserves ''Π''. The label ''representation'' is sometimes also applied to the homomorphism itself (such as
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
).


Representation theory

Perhaps the most well-developed example of this general notion is the subfield of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
called
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, which studies the representing of elements of
algebraic structures In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
by linear transformations of vector spaces.


Other examples

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.


Graph theory

An active area of
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of sets (or, more precisely, non-disjointness) is a
symmetric relation A symmetric relation is a type of binary relation. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: : \forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation ''aRb'' means that . An example is the relation "is equ ...
. This gives rise to the study of intersection graphs for innumerable families of sets. One foundational result here, due to Paul Erdős and his colleagues, is that every ''n''- vertex graph may be represented in terms of intersection among
subsets In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subse ...
of a set of size no more than ''n''2/4. Representing a graph by such algebraic structures as its
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ...
and Laplacian matrix gives rise to the field of
spectral graph theory In mathematics, spectral graph theory is the study of the properties of a Graph (discrete mathematics), graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacen ...
.


Order theory

Dual to the observation above that every graph is an intersection graph is the fact that every
partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
(also known as poset) is isomorphic to a collection of sets ordered by the inclusion (or containment) relation ⊆. Some posets that arise as the
inclusion order In the mathematical field of order theory, an inclusion order is the partial order that arises as the subset-inclusion relation on some collection of objects. In a simple way, every poset ''P'' = (''X'',≤) is (isomorphic to) an inclusion orde ...
s for natural classes of objects include the Boolean lattices and the orders of dimension ''n''. 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 graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
s, as those graphs whose vertex-edge incidence relations are circle orders. There are also geometric representations that are not based on inclusion. Indeed, one of the best studied classes among these are the interval orders, which represent the partial order in terms of what might be called ''disjoint precedence'' of intervals on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
: each element ''x'' of the poset is represented by an interval 'x''1, ''x''2 such that for any ''y'' and ''z'' in the poset, ''y'' is below ''z'' if and only if ''y''2 < ''z''1.


Logic

In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, the representability of algebras as relational structures is often used to prove the equivalence of algebraic and relational semantics. Examples of this include Stone's representation of
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
s as fields of sets, Esakia's representation of Heyting algebras as Heyting algebras of sets, and the study of representable relation algebras and representable cylindric algebras.


Polysemy

Under certain circumstances, a single function ''f'' : ''X'' → ''Y'' is at once an isomorphism from several mathematical structures on ''X''. Since each of those structures may be thought of, intuitively, as a meaning of the image ''Y'' (one of the things that ''Y'' is trying to tell us), this phenomenon is called polysemy—a term borrowed from linguistics. Some examples of polysemy include: * intersection polysemy—pairs of graphs ''G''1 and ''G''2 on a common vertex set ''V'' that can be simultaneously represented by a single collection of sets ''Sv'', such that any distinct vertices ''u'' and ''w'' in ''V'' are adjacent in ''G''1, if and only if their corresponding sets intersect ( ''Su'' ∩ ''Sw'' ≠ Ø ), and are adjacent in ''G''2 if and only if the complements do ( ''Su''C ∩ ''Sw''C ≠ Ø ). * competition polysemy—motivated by the study of ecological
food web A food web is the natural interconnection of food chains and a graphical representation of what-eats-what in an ecological community. Position in the food web, or trophic level, is used in ecology to broadly classify organisms as autotrophs or he ...
s, in which pairs of species may have prey in common or have predators in common. A pair of graphs ''G''1 and ''G''2 on one vertex set is competition polysemic, if and only if there exists a single directed graph ''D'' on the same vertex set, such that any distinct vertices ''u'' and ''v'' are adjacent in ''G''1, if and only if there is a vertex ''w'' such that both ''uw'' and ''vw'' are arcs in ''D'', and are adjacent in ''G''2, if and only if there is a vertex ''w'' such that both ''wu'' and ''wv'' are arcs in ''D''. * interval polysemy—pairs of posets ''P''1 and ''P''2 on a common ground set that can be simultaneously represented by a single collection of real intervals, that is an interval-order representation of ''P''1 and an interval-containment representation of ''P''2.


See also

*
Group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
* Representation theorems *
Model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...


References

{{Reflist Mathematical relations