Noncommutative algebras

Singular.jl allows the creation of various noncommutative algebras over any of the coefficient rings described above. The constructors of the parent objects and elements thereof are given in the following table.

ConstructorElement typeParent typeSINGULAR kernel subsystem
GAlgebraspluralg{T}PluralRing{T}PLURAL
WeylAlgebraspluralg{T}PluralRing{T}PLURAL
FreeAlgebraslpalg{T}LPRing{T}LETTERPLACE

These types are parameterized by the type of elements in the coefficient ring of the algebra. All noncommutative algebra element types belong directly to the abstract type AbstractAlgebra.NCRingElem and all the noncommuative algebra parent object types belong to the abstract type AbstractAlgebra.NCRing. The following union types cover all of Singular polynomial rings and algebras.

const PolyRingUnion{T} = Union{PolyRing{T}, PluralRing{T}, LPRing{T}} where T <: Nemo.RingElem

const SPolyUnion{T} = Union{spoly{T}, spluralg{T}, slpalg{T}} where T <: Nemo.RingElem

Constructors

All constructors returns a tuple, $R, x$ consisting of a parent object $R$ and an array $x$ of variables from which elements of the algebra can be constructed.

For constructors taking an ordering, two orderings can be specified by symbol, one for term ordering, and a second one for ordering of module components. The first ordering can also be specified by a non-symbol as with PolynomialRing, in which case the second ordering is ignored.

By default there will only be one parent object in the system for each combination of arguments. This is accomplished by making use of a global cache. If this is not the desired behaviour cached = false may be passed.

GAlgebra

GAlgebra(R::PolyRing{T}, C::smatrix{spoly{T}}, D::smatrix{spoly{T}};
         cached::Bool = true) where T <: Nemo.RingElem

Construct the G-algebra from a commutative polynomial ring $R$ and matrices $C$, $D$ over $R$. If the variables of $R$ are $x_1,\dots,x_n$, then the noncommutative algebra is constructed with relations $x_j x_i = c_{i,j} x_i x_j + d_{i,j}$ for $1 \le i < j \le n$. The $c_{i,j}$ must be nonzero constant polynomials and the relations $x_i x_j > \mathrm{lm}(d_{i,j})$ must hold in the monomial ordering of the ring $R$.

The entries of the matrices $C$ and $D$ on or below the main diagonal are ignored. A non-matrix argument a for either C or D is turned into a matrix with all relevant entries set to a.

Note

The conditions that assure that multiplication is associative in the resulting algebra are currently not checked. The example below illustrates how this condition can be checked.

Examples

julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"]);

julia> G, (x, y) = GAlgebra(R, 2, Singular.Matrix(R, [0 x; 0 0]))
(Singular G-Algebra (QQ),(x,y),(dp(2),C), spluralg{n_Q}[x, y])

julia> y*x
2*x*y + x

Associativity can be checked via Interpreter Functionality.

julia> iszero(Singular.LibNctools.ndcond(G))
true

The construction of a GR-algebra proceeds by taking the quotient of a G-algebra by a two-sided ideal. Continuing with the above example:

julia> I = Ideal(G, [x^2 + y^2], twosided = true)
Singular two-sided ideal over Singular G-Algebra (QQ),(x,y),(dp(2),C) with generators (x^2 + y^2)

julia> Q, (x, y) = QuotientRing(G, std(I))
(Singular G-Algebra Quotient Ring (QQ),(x,y),(dp(2),C), spluralg{n_Q}[x, y])

WeylAlgebra

function WeylAlgebra(R::Union{Ring, Field}, s::Union{Vector{String}, Vector{Symbol}};
                     ordering = :degrevlex, ordering2::Symbol = :comp1min,
                     cached::Bool = true, degree_bound::Int = 0)

function WeylAlgebra(R::Union{Ring, Field}, s::Union{Matrix{String}, Matrix{Symbol}};
                     ordering = :degrevlex, ordering2::Symbol = :comp1min,
                     cached::Bool = true, degree_bound::Int = 0)

Construct the ring of differential operators $\partial_1, \dots, \partial_n$ with coefficients in $R[x_1, \dots, x_n]$. In the first variant, the differential operators are named by simply appending the letter "d" to each of the strings in x. The second variant takes the names of the $x_i$ from the first row of the matrix and the names of the $\partial_i$ from the second row of the matrix. Note that the functionality of this constructor can be achieved with the GAlgebra constructor: it is provided only for convenience. Note also that due to the ordering constraint on G-algebras, the orderings :neglex, :negdeglex, :negdevrevlex are excluded.

Examples

julia> R, (x, y, dx, dy) = WeylAlgebra(ZZ, ["x", "y"])
(Singular G-Algebra (ZZ),(x,y,dx,dy),(dp(4),C), spluralg{n_Z}[x, y, dx, dy])

julia> (dx*x, dx*y, dy*x, dy*y)
(x*dx + 1, y*dx, x*dy, y*dy + 1)

The ideals of G-algebras are left ideals by default.

julia> R, (x1, x2, x3, d1, d2, d3) = WeylAlgebra(QQ, ["x1" "x2" "x3"; "d1" "d2" "d3"])
(Singular G-Algebra (QQ),(x1,x2,x3,d1,d2,d3),(dp(6),C), spluralg{n_Q}[x1, x2, x3, d1, d2, d3])

julia> gens(std(Ideal(R, [x1^2*d2^2 + x2^2*d3^2, x1*d2 + x3])))
7-element Vector{spluralg{n_Q}}:
 x1*d2 + x3
 x3^2
 x2*x3 - x1
 x1*x3
 x2^2
 x1*x2
 x1^2

FreeAlgebra

FreeAlgebra(R::Field, s::Union{Vector{String}, Vector{Symbol}}, degree_bound::Int;
            ordering = :degrevlex, ordering2::Symbol = :comp1min,
            cached::Bool = true)

Construct the free associative algebra $R \langle x_1,\dots,x_n \rangle$. The ordering must be global.

Note

Since this uses the LETTERPLACE backend, the degree_bound, which is the maximum length on any monomial word in the algebra, must be specified. Multiplication is checked and throws when the resulting degree exceeds this bound.

Examples

julia> R, (x, y) = FreeAlgebra(QQ, ["x", "y"], 5)
(Singular letterplace Ring (QQ),(x,y,x,y,x,y,x,y,x,y),(dp(10),C,L(3)), slpalg{n_Q}[x, y])

julia> (x*y)^2
x*y*x*y

julia> (x*y)^3
ERROR: degree bound of Letterplace ring is 5, but at least 6 is needed for this multiplication

The ideals are two-sided by default for this algebra, and there is currently no possibility of constructing one-sided ideals.

julia> R, (x, y, z) = FreeAlgebra(QQ, ["x", "y", "z"], 4)
(Singular letterplace Ring (QQ),(x,y,z,x,y,z,x,y,z,x,y,z),(dp(12),C,L(3)), slpalg{n_Q}[x, y, z])

julia> gens(std(Ideal(R, [x*y + y*z, x*x + x*y - y*x - y*y])))
8-element Vector{slpalg{n_Q}}:
 x*y + y*z
 x^2 - y*x - y^2 - y*z
 y^3 + y*z*y - y^2*z - y*z^2
 y^2*x + y*z*x + y^2*z + y*z^2
 y^2*z*y + y*z^2*y - y^2*z^2 - y*z^3
 y*z*y^2 + y*z^2*y - y*z*y*z - y*z^3
 y^2*z*x + y*z^2*x + y^2*z^2 + y*z^3
 y*z*y*x + y*z^2*x + y*z*y*z + y*z^3

Term Iterators

For GAlgebra and WeylAlgebra, the elements can be and are represented using commutative data structures, and the function exponent_vectors is repurposed for access to the individual exponents.

Examples

julia> R, (x, y, dx, dy) = WeylAlgebra(QQ, ["x", "y"])
(Singular G-Algebra (QQ),(x,y,dx,dy),(dp(4),C), spluralg{n_Q}[x, y, dx, dy])

julia> p = (dx + dy)*(x + y)
x*dx + y*dx + x*dy + y*dy + 2

julia> show(collect(exponent_vectors(p)))
[[1, 0, 1, 0], [0, 1, 1, 0], [1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 0, 0]]

For FreeAlgebra, the function exponent_vectors is undefined on elements and replaced by exponent_words which reads off in order the indices of the variables in a monomial. Also, the monomials for the MPolyBuildCtx are specified by these exponent words. Other than these differences the term iterators have the same behavior as in the commutative case.

Examples

julia> R, (x, y, z) = FreeAlgebra(QQ, ["x", "y", "z"], 6)
(Singular letterplace Ring (QQ),(x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z),(dp(18),C,L(3)), slpalg{n_Q}[x, y, z])

julia> p = (1 + x*z + y)^2
x*z*x*z + x*z*y + y*x*z + y^2 + 2*x*z + 2*y + 1

julia> show(collect(coefficients(p)))
n_Q[1, 1, 1, 1, 2, 2, 1]

julia> show(collect(monomials(p)))
slpalg{n_Q}[x*z*x*z, x*z*y, y*x*z, y^2, x*z, y, 1]

julia> show(collect(terms(p)))
slpalg{n_Q}[x*z*x*z, x*z*y, y*x*z, y^2, 2*x*z, 2*y, 1]

julia> show(collect(exponent_words(p)))
[[1, 3, 1, 3], [1, 3, 2], [2, 1, 3], [2, 2], [1, 3], [2], Int64[]]

julia> B = MPolyBuildCtx(R)
Builder for a polynomial in Singular letterplace Ring (QQ),(x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z),(dp(18),C,L(3))

julia> push_term!(B, QQ(2), [3,2,1,3]);

julia> push_term!(B, QQ(-1), Int[]);

julia> finish(B)
2*z*y*x*z - 1