Missing preimage methods and/or need to extend the map specification

[fingolfin]

There are two tests in AA which seem to use preimage:

test/generic/QuotientModule-test.jl:177:         @test h(preimage(h, n)) == n
test/generic/Submodule-test.jl:160:         pre = preimage(f, n)

But in general there don't seem to be preimage methods. I took this examples fromtest/generic/MapWithInverse-test.jl:

julia> u = map_with_section_from_func(x -> x + 1, x -> x - 1, ZZ, ZZ)
Map with section
  from integers
  to integers

julia> u(ZZ(1))
2

But playing with this reveals image and preimage don't exist and in general I found no way to get a preimage other than asking for the preimage map

julia> image(u, ZZ(1))
ERROR: MethodError: no method matching image(::AbstractAlgebra.Generic.MapWithSection{AbstractAlgebra.Integers{BigInt}, AbstractAlgebra.Integers{BigInt}}, ::BigInt)
Stacktrace:
 [1] top-level scope
   @ REPL[34]:1

julia> preimage(u, ZZ(1))
ERROR: MethodError: no method matching preimage(::AbstractAlgebra.Generic.MapWithSection{AbstractAlgebra.Integers{BigInt}, AbstractAlgebra.Integers{BigInt}}, ::BigInt)

Closest candidates are:
  preimage(::AbstractAlgebra.Generic.CompositeMap, ::Any)
   @ AbstractAlgebra ~/Projekte/OSCAR/AbstractAlgebra.jl/src/NemoStuff.jl:564

Stacktrace:
 [1] top-level scope
   @ REPL[35]:1

We only have this:

julia> preimage_map(u)
Map defined by a Julia function
  from integers
  to integers

IMHO preimage should be supported by all maps where it makes sense (whether they have a section or are outright invertible). And personally I'd also like to see image(f,x) work in addition to f(x) but that's secondary.

Originally posted by @fingolfin in #1441 (comment)