# Category: Morphisms

Morphism of finite type
For a homomorphism A → B of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means tha
Graph homomorphism
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs t
Diagonal morphism
In category theory, a branch of mathematics, for any object in any category where the product exists, there exists the diagonal morphism satisfying for where is the canonical projection morphism to th
Antiisomorphism
In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets A and B is an isomorphism from A to the opposite of B (or equivalently from the opposite o
Automorphism of a Lie algebra
In abstract algebra, an automorphism of a Lie algebra is an isomorphism between and itself; i.e., a linear automorphism that preserves the bracket. The totality of them forms the automorphism group of
Catamorphism
In category theory, the concept of catamorphism (from the Ancient Greek: κατά "downwards" and μορφή "form, shape") denotes the unique homomorphism from an initial algebra into some other algebra. In f
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all o
Normal morphism
In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism.A normal category is a category in which every mon
Normal homomorphism
In algebra, a normal homomorphism is a ring homomorphism that is flat and is such that for every field extension L of the residue field of any prime ideal , is a normal ring.
Algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if A and B are algebras over a field (or commutative ring) K, it is a function such that for
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an is
Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
Homomorphism
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 word homomorphism comes from the A
In algebra, an additive map, -linear map or additive function is a function that preserves the addition operation: for every pair of elements and in the domain of For example, any linear map is additi
Ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f
Graph isomorphism
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H such that any two vertices u and v of G are adjacent in G if and only if and are adjacent in H. This
Zero morphism
In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.
Finite morphism
In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over . This defini
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of cont
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of cat
Graph isomorphism problem
Unsolved problem in computer science: Can the graph isomorphism problem be solved in polynomial time? (more unsolved problems in computer science) The graph isomorphism problem is the computational pr
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 where the group operation on the le
Epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1,
Antihomomorphism
In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an an
Strong monomorphism
No description available.
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is
Unramified morphism
In algebraic geometry, an unramified morphism is a morphism of schemes such that (a) it is locally of finite presentation and (b) for each and , we have that 1. * The residue field is a separable alg
Order isomorphism
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever t