Submodules
AbstractAlgebra allows the construction of submodules/subvector spaces of AbstractAlgebra modules over euclidean domains. These are given as the submodule generated by a finite list of elements in the original module.
We define two submodules to be equal if they are (transitively) submodules of the same module $M$ and their generators generate the same set of elements.
Generic submodule type
AbstractAlgebra implements a generic submodule type Generic.Submodule{T}
where T
is the element type of the base ring in src/generic/Submodule.jl
. See src/generic/GenericTypes.jl
for more details of the type definition.
Elements of a generic submodule have type Generic.SubmoduleElem{T}
.
Abstract types
Submodule types belong to the abstract type FPModule{T}
and their elements to FPModuleElem{T}
.
Constructors
AbstractAlgebra.sub
— Methodsub(m::FPModule{T}, gens::Vector{<:FPModuleElem{T}}) where T <: RingElement
Return the submodule of the module m
generated by the given generators, given as elements of m
.
AbstractAlgebra.sub
— Methodsub(m::Module{T}, subs::Vector{<:Generic.Submodule{T}}) where T <: RingElement
Return the submodule S
of the module m
generated by the union of the given submodules of $m$, and a map which is the canonical injection from S
to m
.
Note that the preimage of the canonical injection can be obtained using the preimage function described in the section on module homomorphisms. As the canonical injection is injective, this is unique.
Examples
julia> M = free_module(ZZ, 2)
Free module of rank 2 over integers
julia> m = M([ZZ(1), ZZ(2)])
(1, 2)
julia> n = M([ZZ(2), ZZ(-1)])
(2, -1)
julia> N, f = sub(M, [m, n])
(Submodule over integers with 2 generators and no relations, Hom: N -> M)
julia> v = N([ZZ(3), ZZ(4)])
(3, 4)
julia> v2 = f(v)
(3, 26)
julia> V = vector_space(QQ, 2)
Vector space of dimension 2 over rationals
julia> m = V([QQ(1), QQ(2)])
(1//1, 2//1)
julia> n = V([QQ(2), QQ(-1)])
(2//1, -1//1)
julia> N, f = sub(V, [m, n])
(Subspace over rationals with 2 generators and no relations, Hom: N -> V)
Functionality for submodules
In addition to the Module interface, AbstractAlgebra submodules implement the following functionality.
Basic manipulation
AbstractAlgebra.Generic.supermodule
— Methodsupermodule(M::Submodule{T}) where T <: RingElement
Return the module that this module is a submodule of.
AbstractAlgebra.Generic.is_submodule
— Methodis_submodule(M::AbstractAlgebra.FPModule{T}, N::AbstractAlgebra.FPModule{T}) where T <: RingElement
Return true
if $N$ was constructed as a submodule of $M$. The relation is taken transitively (i.e. subsubmodules are submodules for the purposes of this relation, etc). The module $M$ is also considered a submodule of itself for this relation.
AbstractAlgebra.Generic.is_compatible
— Methodis_compatible(M::AbstractAlgebra.FPModule{T}, N::AbstractAlgebra.FPModule{T}) where T <: RingElement
Return true, P
if the given modules are compatible, i.e. that they are (transitively) submodules of the same module, P. Otherwise return false, M
.
AbstractAlgebra.Generic.dim
— Methoddim(N::Submodule{T}) where T <: FieldElement
Return the dimension of the given vector subspace.
Examples
julia> M = free_module(ZZ, 2)
Free module of rank 2 over integers
julia> m = M([ZZ(2), ZZ(3)])
(2, 3)
julia> n = M([ZZ(1), ZZ(4)])
(1, 4)
julia> N1, = sub(M, [m, n])
(Submodule over integers with 2 generators and no relations, Hom: N1 -> M)
julia> N2, = sub(M, [m])
(Submodule over integers with 1 generator and no relations, Hom: N2 -> M)
julia> supermodule(N1) == M
true
julia> is_compatible(N1, N2)
(true, Free module of rank 2 over integers)
julia> is_submodule(N1, M)
false
julia> V = vector_space(QQ, 2)
Vector space of dimension 2 over rationals
julia> m = V([QQ(2), QQ(3)])
(2//1, 3//1)
julia> N, = sub(V, [m])
(Subspace over rationals with 1 generator and no relations, Hom: N -> V)
julia> dim(V)
2
julia> dim(N)
1
Intersection
Base.intersect
— Methodintersect(M::FPModule{T}, N::FPModule{T}) where T <: RingElement
Return the intersection of the modules $M$ as a submodule of $M$. Note that $M$ and $N$ must be (constructed as) submodules (transitively) of some common module $P$.
Examples
julia> M = free_module(ZZ, 2)
Free module of rank 2 over integers
julia> m = M([ZZ(2), ZZ(3)])
(2, 3)
julia> n = M([ZZ(1), ZZ(4)])
(1, 4)
julia> N1 = sub(M, [m, n])
(Submodule over integers with 2 generators and no relations, Hom: submodule over integers with 2 generators and no relations -> M)
julia> N2 = sub(M, [m])
(Submodule over integers with 1 generator and no relations, Hom: submodule over integers with 1 generator and no relations -> M)
julia> I = intersect(N1, N2)
Any[]