Structure Operations

Contents

Related Structures

The main structure related to a free algebra is its coefficient ring. Multivariate free algebras belong to the Magma category AlgFr.

Category(F) : AlgFr -> Cat
Parent(F) : AlgFr -> Pow
PrimeRing(F) : AlgFr -> Rng
BaseRing(F) : AlgFr -> Rng
CoefficientRing(F) : AlgFr -> Rng
Return the coefficient ring of the free algebra F.

Numerical Invariants

Note that the # operator only returns a value for finite (quotients of) free algebras.

Characteristic(F) : AlgFr -> RngIntElt
# F : AlgFr -> RngIntElt
Rank(F) : AlgFr -> RngIntElt
Return the number of indeterminates of free algebra F over its coefficient ring.

Homomorphisms

In its most general form, a homomorphism taking a free algebra K< x1, ..., xn > as domain requires n + 1 pieces of information, namely, a map (homomorphism) telling how to map the coefficient ring K together with the images of the n indeterminates. The map for the coefficient ring is optional.

hom< F -> S | f, y1, ..., yn > : AlgFr, Rng -> Map
hom< F -> S | y1, ..., yn > : AlgFr, Rng -> Map
Given a free algebra F=K< x1, ..., xn >, a ring or associative algebra S (including another FP-algebra or a matrix algebra), and a map f : K -> S and n elements y1, ..., yn∈S, create the homomorphism g : F -> S by applying the rules that g(rx1a1 ... xnan)=f(r)y1a1 ... ynan for monomials and linearity, that is, g(M + N)=g(M) + g(N).

The coefficient ring map may be omitted, in which case the coefficients are mapped into S by the coercion map.

No attempt is made to check whether the map defines a genuine homomorphism.

Example AlgFP_Homomorphism (H91E1)

In this example we map an algebra F first into F itself, and then into a matrix algebra. 
> K := RationalField();
> F<x,y,z> := FreeAlgebra(K, 3);
> h := hom<F -> F | x*y, y*x, z*x>;
> h(x);
x*y
> h(y);
y*x
> h(x*y);
x*y^2*x
> h(x + y + z);
x*y + y*x + z*x
> A := MatrixAlgebra(K, 2);
> M := [A | [1,1,-1,1], [-1,3,4,1], [11,7,-7,8]];
> M;
[
    [ 1  1]
    [-1  1],
    [-1  3]
    [ 4  1],
    [11  7]
    [-7  8]
]
> h := hom<F -> A | M>;
> h(x);
[ 1  1]
[-1  1]
> h(y);
[-1  3]
[ 4  1]
> h(x*y - y*z);
[ 35 -13]
[-32 -38]
V2.29, 21 October 2025