Definition & Betydelse | Engelska ordet HOMOMORPHISM

Definition av HOMOMORPHISM

1. (matematik) homomorfi

Exempel på hur man kan använda HOMOMORPHISM i en mening

• In an algebraic structure such as a group, a ring, or vector space, an automorphism is simply a bijective homomorphism of an object into itself.
• In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A.
• In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
• In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure.
• In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
• The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.
• However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism.
• In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1).
• Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism.
• In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism.
• Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism.
• showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an embedding).
• In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.
• There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup.
• If G and H are two partially ordered groups, a map from G to H is a morphism of partially ordered groups if it is both a group homomorphism and a monotonic function.
• Open mapping theorem (topological groups), states that a surjective continuous homomorphism of a locally compact Hausdorff group G onto a locally compact Hausdorff group H is an open mapping if G is σ-compact.
• The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects.
• For smooth manifolds, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over.
• Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups.
• An equivalent algebraic approach starts from the observation that a supermanifold is determined by its ring of supercommutative smooth functions, and that a morphism of supermanifolds corresponds one to one with an algebra homomorphism between their functions in the opposite direction, i.

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