Representation theorem

Last updated February 25, 2023

In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.

Examples

Algebra

Cayley's theorem states that every group is isomorphic to a permutation group.^[1]
Representation theory studies properties of abstract groups via their representations as linear transformations of vector spaces.
Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets.^[2]
A variant, Stone's representation theorem for distributive lattices, states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.
Another variant, Stone's duality, states that there exists a duality (in the sense of an arrow-reversing equivalence) between the categories of Boolean algebras and that of Stone spaces.
The Poincaré–Birkhoff–Witt theorem states that every Lie algebra embeds into the commutator Lie algebra of its universal enveloping algebra.
Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero embeds into the Lie algebra of endomorphisms of some finite-dimensional vector space.
Birkhoff's HSP theorem states that every model of an algebra A is the homomorphic image of a subalgebra of a direct product of copies of A.^[3]
In the study of semigroups, the Wagner–Preston theorem provides a representation of an inverse semigroup S, as a homomorphic image of the set of partial bijections on S, and the semigroup operation given by composition.

Category theory

The Yoneda lemma provides a full and faithful limit-preserving embedding of any category into a category of presheaves.
Mitchell's embedding theorem for abelian categories realises every small abelian category as a full (and exactly embedded) subcategory of a category of modules over some ring.^[4]
Mostowski's collapsing theorem states that every well-founded extensional structure is isomorphic to a transitive set with the ∈-relation.
One of the fundamental theorems in sheaf theory states that every sheaf over a topological space can be thought of as a sheaf of sections of some (étalé) bundle over that space: the categories of sheaves on a topological space and that of étalé spaces over it are equivalent, where the equivalence is given by the functor that sends a bundle to its sheaf of (local) sections.

Functional analysis

The Gelfand–Naimark–Segal construction embeds any C*-algebra in an algebra of bounded operators on some Hilbert space.
The Gelfand representation (also known as the commutative Gelfand–Naimark theorem) states that any commutative C*-algebra is isomorphic to an algebra of continuous functions on its Gelfand spectrum. It can also be seen as the construction as a duality between the category of commutative C*-algebras and that of compact Hausdorff spaces.
The Riesz–Markov–Kakutani representation theorem is actually a list of several theorems; one of them identifies the dual space of C₀(X) with the set of regular measures on X.

Geometry

The Whitney embedding theorems embed any abstract manifold in some Euclidean space.
The Nash embedding theorem embeds an abstract Riemannian manifold isometrically in a Euclidean space.^[5]

Economics

A preference representation theorem states conditions for the existence of a utility function representing a preference relation. Examples are Von Neumann–Morgenstern utility theorem and Debreu's representation theorems.

Related Research Articles

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.

In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.

In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra.

In mathematics, the Gelfand representation in functional analysis is either of two things:

In category theory, a branch of mathematics, the opposite category or dual categoryC^op of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, $.$

In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space.

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of $A$ is $B$ , then the dual of $B$ is $A$ . Such involutions sometimes have fixed points, so that the dual of $A$ is $A$ itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.

In mathematics, a space is a set with some added structure.

In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.

Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.

In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ₁ compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ₂ compact elements using a construction based on Kuratowski's free set theorem.

In abstract algebra, a skew lattice is an algebraic structure that is a non-commutative generalization of a lattice. While the term skew lattice can be used to refer to any non-commutative generalization of a lattice, since 1989 it has been used primarily as follows.

In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.

In mathematics, duality theory for distributive lattices provides three different representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras.

Introduction to Lattices and Order is a mathematical textbook on order theory by Brian A. Davey and Hilary Priestley. It was published by the Cambridge University Press in their Cambridge Mathematical Textbooks series in 1990, with a second edition in 2002. The second edition is significantly different in its topics and organization, and was revised to incorporate recent developments in the area, especially in its applications to computer science. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.

References

↑ "Cayley's Theorem and its Proof". www.sjsu.edu. Retrieved 2019-12-08.
↑ Dirks, Matthew. "The Stone Representation Theorem for Boolean Algebras" (PDF). math.uchicago.edu. Retrieved 2019-12-08.{{cite web}}: CS1 maint: url-status (link)
↑ Schneider, Friedrich Martin (November 2017). "A uniform Birkhoff theorem". Algebra Universalis. 78 (3): 337–354. arXiv: 1510.03166 . doi:10.1007/s00012-017-0460-1. ISSN 0002-5240.
↑ Freyd–Mitchell embedding theorem at the nLab
↑ "Notes on the Nash embedding theorem". What's new. 2016-05-11. Retrieved 2019-12-08.

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[1] "Cayley's Theorem and its Proof". www.sjsu.edu. Retrieved 2019-12-08.

[2] Dirks, Matthew. "The Stone Representation Theorem for Boolean Algebras" (PDF). math.uchicago.edu. Retrieved 2019-12-08.{{cite web}}: CS1 maint: url-status (link)

[3] Schneider, Friedrich Martin (November 2017). "A uniform Birkhoff theorem". Algebra Universalis. 78 (3): 337–354. arXiv: 1510.03166 . doi:10.1007/s00012-017-0460-1. ISSN 0002-5240.

[4] Freyd–Mitchell embedding theorem at the nLab

[5] "Notes on the Nash embedding theorem". What's new. 2016-05-11. Retrieved 2019-12-08.

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