Representation theorem

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. Cayley's theorem states that every group is isomorphic to a permutation group. 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. 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 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. 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. The Yoneda lemma provides a full and faithful -preserving embedding of any category into a category of presheaves. Mitchell's embedding theorem for abelian categories realises every abelian category as a full (and exactly embedded) of a over some ring. Mostowski's collapsing theorem states that every well-founded extensional structure is isomorphic to a transitive set with the ∈-relation.