# Homomorphisms

Magma has the capability of creating homomorphisms of basic algebras. A homomorphism of a basic algebra has the type Map. The matrix of the homomorphism $\phi$ is accessed by entering `Matrix(phi)`. The kernel of a homomorphism is returned as a subspace of the vector space of the domain of the algebra. This is a two-sided ideal. There is no special type for ideals. They are subspaces of the underlying vector space of the algebra. They can be generated from any given set of elements of the algebra

The image of a homomorphism is returned as a basic algebra, together with the embedding homomorphism.

A homomorphism from a basic algebra $A$ to a basic algebra $B$ is normally created from a matrix having dimension of $A$ rows and dimension of $B$ columns. Note that Magma does not automatically check to see if the created map is an algebra homomorphism.

## `hom<A -> B | S>: AlgBas, AlgBas, ModMatFldElt -> Map`

The algebra homomorphism from basic algebra $A$ to basic algebra $B$, whose matrix is the matrix S. Given a map $\phi$, a homomorphism of basic algebra, the matrix of that map is recalled with the command `Matrix(phi)`.

## `Kernel(phi): Map -> ModTupFld`

The kernel of the map $\phi$.

## `Image(phi): Map -> AlgBas, Map`

The image of the homomorphism $\phi$ together with the embedding homomorphism of the image into the codomain of $\phi$.

## `IsAlgebraHomomorphism(A, B, psi): AlgBas, AlgBas, Mtrx -> Bool`

Return `true` if the matrix $\psi$ represents a homomorphism from basic algebra A to basic algebra B.

## `X * Y: Map, Map -> Map`

The composition of the maps $X$ and $Y$.

## `IsAlgebraHomomorphism(A, B, psi): AlgBas, AlgBas, Map -> Bool`

## `IsAlgebraHomomorphism(A, B, psi): AlgBasGrpP, AlgBasGrpP, Map -> Bool`

## `IsAlgebraHomomorphism(A, B, psi): AlgBasGrpP, AlgBas, Map -> Bool`

## `IsAlgebraHomomorphism(A, B, psi): AlgBas, AlgBasGrpP, Map -> Bool`

Returns `true`, if the map $\psi$ is a homomorphism of basic algebras

## `IsAlgebraHomomorphism(psi): Map -> Bool`

Returns `true` if the map $\psi$ is a homomorphism of basic algebras.