Direct Sums

Direct Sums

AbstractAlgebra allows the construction of the external direct sum of any nonempty vector of finitely presented modules.

As well as implementing the entire Module interface, AbstractAlgebra direct sums also provide the following interface.

Note that external direct sums are considered equal iff they are the same object.

Constructors

DirectSum(m::Vector{<:Module{T}}) where T <: RingElement

Return a tuple $M, f, g$ consisting of $M$ the direct sum of the modules m (supplied as a vector of modules), a vector $f$ of the canonical injections of the $m[i]$ into $M$ and a vector $g$ of the canonical projections from $M$ onto the $m[i]$.

DirectSum(m::Module{T}...) where T <: RingElement

Return a tuple $M, f, g$ consisting of $M$ the direct sum of the given modules, a vector $f$ of the canonical injections of the $m[i]$ into $M$ and a vector $g$ of the canonical projections from $M$ onto the $m[i]$.

Examples

julia> F = FreeModule(ZZ, 5)
Free module of rank 5 over Integers

julia> m1 = F(BigInt[4, 7, 8, 2, 6])
(4, 7, 8, 2, 6)

julia> m2 = F(BigInt[9, 7, -2, 2, -4])
(9, 7, -2, 2, -4)

julia> S1, f1 = sub(F, [m1, m2])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 5 over Integers)

julia> m1 = F(BigInt[3, 1, 7, 7, -7])
(3, 1, 7, 7, -7)

julia> m2 = F(BigInt[-8, 6, 10, -1, 1])
(-8, 6, 10, -1, 1)

julia> S2, f2 = sub(F, [m1, m2])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 5 over Integers)

julia> m1 = F(BigInt[2, 4, 2, -3, -10])
(2, 4, 2, -3, -10)

julia> m2 = F(BigInt[5, 7, -6, 9, -5])
(5, 7, -6, 9, -5)

julia> S3, f3 = sub(F, [m1, m2])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 5 over Integers)

julia> D, f = DirectSum(S1, S2, S3)
(DirectSumModule over Integers, AbstractAlgebra.Generic.ModuleHomomorphism{BigInt}[Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: DirectSumModule over Integers, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: DirectSumModule over Integers, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: DirectSumModule over Integers], AbstractAlgebra.Generic.ModuleHomomorphism{BigInt}[Module homomorphism with
Domain: DirectSumModule over Integers
Codomain: Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: DirectSumModule over Integers
Codomain: Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: DirectSumModule over Integers
Codomain: Submodule over Integers with 2 generators and no relations
])

Functionality for direct sums

In addition to the Module interface, AbstractAlgebra direct sums implement the following functionality.

Basic manipulation

summands(M::DirectSumModule{T}) where T <: RingElement

Return the modules that this module is a direct sum of.

Examples

julia> F = FreeModule(ZZ, 5)
Free module of rank 5 over Integers

julia> m1 = F(BigInt[4, 7, 8, 2, 6])
(4, 7, 8, 2, 6)

julia> m2 = F(BigInt[9, 7, -2, 2, -4])
(9, 7, -2, 2, -4)

julia> S1, f1 = sub(F, [m1, m2])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 5 over Integers)

julia> m1 = F(BigInt[3, 1, 7, 7, -7])
(3, 1, 7, 7, -7)

julia> m2 = F(BigInt[-8, 6, 10, -1, 1])
(-8, 6, 10, -1, 1)

julia> S2, f2 = sub(F, [m1, m2])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 5 over Integers)

julia> m1 = F(BigInt[2, 4, 2, -3, -10])
(2, 4, 2, -3, -10)

julia> m2 = F(BigInt[5, 7, -6, 9, -5])
(5, 7, -6, 9, -5)

julia> S3, f3 = sub(F, [m1, m2])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 5 over Integers)

julia> D, f = DirectSum(S1, S2, S3)
(DirectSumModule over Integers, AbstractAlgebra.Generic.ModuleHomomorphism{BigInt}[Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: DirectSumModule over Integers, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: DirectSumModule over Integers, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: DirectSumModule over Integers], AbstractAlgebra.Generic.ModuleHomomorphism{BigInt}[Module homomorphism with
Domain: DirectSumModule over Integers
Codomain: Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: DirectSumModule over Integers
Codomain: Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: DirectSumModule over Integers
Codomain: Submodule over Integers with 2 generators and no relations
])

julia> summands(D)
3-element Array{AbstractAlgebra.Generic.Submodule{BigInt},1}:
 Submodule over Integers with 2 generators and no relations

 Submodule over Integers with 2 generators and no relations

 Submodule over Integers with 2 generators and no relations
    (D::DirectSumModule{T}(::Vector{<:AbstractAlgebra.FPModuleElem{T}}) where T <: RingElement

Given a vector (or $1$-dim array) of module elements, where the $i$-th entry has to be an element of the $i$-summand of $D$, create the corresponding element in $D$.

Examples

julia> N = FreeModule(QQ, 2);

julia> M = FreeModule(QQ, 1);

julia> D, _ = DirectSum(M, N, M);

julia> D([gen(M, 1), gen(N, 1), gen(M, 2)])
(1//1, 1//1, 0//1, 0//1)

Special Homomorphisms

Due to the special structure as direct sums, homomorphisms can be created by specifying homomorphisms for all summands. In case of the codmain being a direct sum as well, any homomorphism may be thought of as a matrix containing maps from the $i$-th source summand to the $j$-th target module:

ModuleHomomorphism(D::DirectSumModule{T}, S::DirectSumModule{T}, m::Array{Any, 2}) where T <: RingElement

Given a matrix $m$ such that the $(i,j)$-th entry is either $0$ (Int(0)) or a ModuleHomomorphism from the $i$-th summand of $D$ to the $j$-th summand of $S$, construct the corresponding homomorphism.

ModuleHomomorphism(D::DirectSumModule{T}, S::FPModuleElem{T}, m::Array{ModuleHomomorphism, 1})

Given an array $a$ of ModuleHomomorphism such that $a_i$, the $i$-th entry of $a$ is a ModuleHomomorphism from the $i$-th summand of D into S, construct the direct sum of the components.

Given a matrix $m$ such that the $(i,j)$-th entry is either $0$ (Int(0)) or a ModuleHomomorphism from the $i$-th summand of $D$ to the $j$-th summand of $S$, construct the corresponding homomorphism.

Examples

julia> N = FreeModule(QQ, 2);

julia> D, _ = DirectSum(N, N);

julia> p = ModuleHomomorphism(N, N, [3,4] .* basis(N));

julia> q = ModuleHomomorphism(N, N, [5,7] .* basis(N));

julia> phi = ModuleHomomorphism(D, D, [p 0; 0 q])
Module homomorphism with
Domain: DirectSumModule over Rationals
Codomain: DirectSumModule over Rationals

julia> r = ModuleHomomorphism(N, D, [2,3] .* gens(D)[1:2])
Module homomorphism with
Domain: Vector space of dimension 2 over Rationals
Codomain: DirectSumModule over Rationals

julia> psi = ModuleHomomorphism(D, D, [r, r])
Module homomorphism with
Domain: DirectSumModule over Rationals
Codomain: DirectSumModule over Rationals