Operations on Mappings

Composition

Although compatible maps can be composed by repeated application, say \(g(f(x))\), it is also possible to create a composite map.

f * g: Map, Map -> Map

Given a mapping \(f : A \rightarrow B\), and a mapping \(g : B \rightarrow C\), construct the composition \(h\) of the mappings \(f\) and \(g\) as the mapping \(h=g\circ f : A \rightarrow C\).

Components(f): Map -> [ Map ]

Returns the maps which were composed to form \(f\).

(Co)Domain and (Co)Kernel

The domain and codomain of any map can simply be accessed. Only for some intrinsic maps and for maps with certain domains and codomains, also the formation of image, kernel, and cokernel is available.

Domain(f): Map -> Str

The domain of the mapping \(f\).

Codomain(f): Map -> Str

The codomain of the mapping \(f\).

Image(f): Map -> Elt

Given a mapping \(f\) with domain \(A\) and codomain \(B\), return the image of \(A\) in \(B\) as a substructure of \(B\). This function is currently supported only for some intrinsic maps and for maps with certain domains and codomains.

Kernel(f): Map -> Str

Given the homomorphism \(f\) with domain \(A\) and codomain \(B\), return the kernel of \(f\) as a substructure of \(A\). This function is currently supported only for some intrinsic maps and for maps with certain domains and codomains.

Inverse

Inverse(m): Map -> Map

The inverse map of the map \(m\).

Function

For a map given by a rule, it is possible to get access to the rule as a user defined function.

Function(f): Map -> UserProgram

The function underlying the mapping \(f\). Only available if \(f\) has been defined by the user by means of a rule map (i.e., an expression for the image under \(f\) of an arbitrary element of the domain).