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Map with inverse

It is not possible to provide generic functionality to invert a map. However, sometimes one knows an inverse map explicitly and would like to keep track of this.

Recall that as map composition is not commutative, there is a notion of a left inverse and a right inverse for maps.

To keep track of such inverse maps, AbstractAlgebra provides data types Generic.MapWithRetraction and Generic.MapWithSection.

Given a map $f : X \to Y$, a retraction of $f$ is a map $g : Y \to X$ such that $g(f(x)) = x$ for all $x \in X$.

Given a map $f : X \to Y$, a section of $f$ is a map $g : Y \to X$ such that $f(g(x)) = x$ for all $y \in Y$.

In AbstractAlgebra, a map with retraction/section is an object containing a pair of maps, the second of which is a retraction/section of the first.

Maps with retraction/section can be composed, and we also define the inverse of such a pair to be the map with the pair swapped. Thus the inverse of a map with retraction is a map with section.

Map with inverse constructors

To construct a map with retraction/section from a pair of maps, we have the following functions:

AbstractAlgebra.map_with_retraction — Function
map_with_retraction(f::Map{D, C}, r::Map{C, D}) where {D, C}

Return the map f together with a known retraction r of it, i.e. a map with $r(f(x)) = x$ for all $x$ in the domain of f.

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AbstractAlgebra.map_with_section — Function
map_with_section(f::Map{D, C}, s::Map{C, D}) where {D, C}

Return the map f together with a known section s of it, i.e. a map with $f(s(y)) = y$ for all $y$ in the codomain of f.

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For convenience we allow construction of maps with retraction/section from a pair of Julia functions/closures.

AbstractAlgebra.map_with_retraction_from_func — Function
map_with_retraction_from_func(f::Function, r::Function, R, S)

Return the map from R to S given by the Julia function f, together with the retraction given by the Julia function r. See map_with_retraction.

Examples

julia> f = map_with_retraction_from_func(x -> x + 1, x -> x - 1, ZZ, ZZ)
Map with retraction
  from integers
  to integers

julia> f(ZZ(1))
2
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Functionality for maps with inverses

The following functionality is provided for maps with inverses.

inv(M::Generic.MapWithRetraction)
inv(M::Generic.MapWithSection)

Return the map with the two maps contained in $M$ swapped. In the first case, a MapWithSection is returned. In the second case a MapWithRetraction is returned.

Examples

julia> f = map_with_retraction_from_func(x -> x + 1, x -> x - 1, ZZ, ZZ)
Map with retraction
  from integers
  to integers

julia> g = inv(f)
Map with section
  from integers
  to integers

julia> h = f*g
Composite map
  from integers
  to integers
which is the composite of
  Map: integers -> integers
  Map: integers -> integers

julia> h(ZZ(1))
1

To access the two maps stored in a map with retraction/section, we have the following:

AbstractAlgebra.Generic.image_map — Function
image_map(f::Union{Generic.MapWithSection, Generic.MapWithRetraction})

Return the underlying map of f, i.e. f stripped of its section resp. retraction.

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