Arithmetic with Words

Construction of a Word

Identity(G): GrpRWS -> GrpRWSElt
Id(G): GrpRWS -> GrpRWSElt
G ! 1: GrpRWS, RngIntElt -> GrpRWSElt

Construct the identity word in \(G\).

G ! [ i₁, ..., iₛ ]: GrpRWS, [ RngIntElt ] -> GrpRWSElt

Given a rewrite group \(G\) defined on \(r\) generators and a sequence \([i_1, \cdots, i_s]\) of integers lying in the range \([-r, r]\), excluding \(0\), construct the word

\[G.|i_1|^{\epsilon_1}*G.|i_2|^{\epsilon_2}* \cdots *G.|i_s|^{\epsilon_s}\]

where \(\epsilon_j\) is \(+1\) if \(i_j\) is positive, and \(-1\) if \(i_j\) is negative. The resulting word is reduced using the reduction machine associated with \(G\).

Parent(w): GrpRWSElt -> GrpRWS

The parent group \(G\) for the word \(w\).

Example: Constructing Words in a Rewrite Group (ex-6e6f68)

We construct the Fibonacci group \(F(2,7)\), and its identity.

> FG<a,b,c,d,e,f,g> := FreeGroup(7);
> F := quo< FG | a*b=c, b*c=d, c*d=e, d*e=f, e*f=g, f*g=a, g*a=b>;
> G := RWSGroup(F : TidyInt := 1000);
> Id(G);
Id(G)
> G!1;
Id(G)
> G![1,2];
G.3

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Element Operations

Having constructed a rewrite group \(G\) one can perform arithmetic with words in \(G\). Assuming we have \(u, v \in G\) then the product \(u*v\) will be computed as follows:

(i)

the product \(w = u*v\) is formed as a product in the appropriate free group.

(ii)

\(w\) is reduced using the reduction machine associated with \(G\).

If \(G\) is confluent, then \(w\) will be the unique minimal word that represents \(u*v\) under the ordering of \(G\). If \(G\) is not confluent, then there are some pairs of words which are equal in \(G\), but which reduce to distinct words, and hence \(w\) will not be a unique normal form. Note that:

(i)

reduction of \(w\) can cause an increase in the length of \(w\). At present there is an internal limit on the length of a word – if this limit is exceeded during reduction an error will be raised. Hence any word operation involving reduction can fail.

(ii)

the implementation is designed more with speed of execution in mind than with minimizing space requirements; thus, the reduction machine is always used to carry out word reduction, which can be space-consuming, particularly when the number of generators is large.

u * v: GrpRWSElt, GrpRWSElt -> GrpRWSElt

Given words \(w\) and \(v\) belonging to a common group, return their product.

u / v: GrpRWSElt, GrpRWSElt -> GrpRWSElt

Given words \(w\) and \(v\) belonging to a common group, return the product of the word \(u\) by the inverse of the word \(v\), i.e. the word \(u*v^{-1}\).

u ^ n: GrpRWSElt, RngIntElt -> GrpRWSElt

The \(n\)-th power of the word \(w\).

u ^ v: GrpRWSElt, GrpRWSElt -> GrpRWSElt

Given words \(w\) and \(v\) belonging to a common group, return the conjugate of the word \(u\) by the word \(v\), i.e. the word \(v^{-1}*u*v\).

Inverse(w): GrpRWSElt -> GrpRWSElt

The inverse of the word \(w\).

(u, v): GrpRWSElt, GrpRWSElt -> GrpRWSElt

Given words \(w\) and \(v\) belonging to a common group, return the commutator of the words \(u\) and \(v\), i.e., the word \(u^{-1}v^{-1}uv\).

(u₁, ..., uᵣ): GrpRWSElt, ..., GrpRWSElt -> GrpRWSElt

Given \(r\) words \(u_1, \ldots, u_r\) belonging to a common group, return their commutator. Commutators are left-normed, so they are evaluated from left to right.

u eq v: GrpRWSElt, GrpRWSElt -> BoolElt

Given words \(w\) and \(v\) belonging to the same group, return true if \(w\) and \(v\) reduce to the same normal form, false otherwise. If \(G\) is confluent this tests for equality. If \(G\) is non-confluent then two words which are the same may not reduce to the same normal form.

u ne v: GrpRWSElt, GrpRWSElt -> BoolElt

Given words \(w\) and \(v\) belonging to the same group, return false if \(w\) and \(v\) reduce to the same normal form, true otherwise. If \(G\) is confluent this tests for non-equality. If \(G\) is non-confluent then two words which are the same may reduce to different normal forms.

IsId(w): GrpRWSElt -> BoolElt
IsIdentity(w): GrpRWSElt -> BoolElt

Returns true if the word \(w\) is the identity word.

# u: GrpRWSElt -> RngIntElt

The length of the word \(w\).

ElementToSequence(u): GrpRWSElt -> [ RngIntElt ]
Eltseq(u): GrpRWSElt -> [ RngIntElt ]

The sequence \(Q\) obtained by decomposing the element \(u\) of a rewrite group into its constituent generators and generator inverses. Suppose \(u\) is a word in the rewrite group \(G\). Then, if \(u = G.i_1^{e_1} \cdots G.i_m^{e_m}\), with each \(e_i = \pm 1\), then \(Q[j] = i_j\) if \(e_j = +1\) and \(Q[j] = -i_j\) if \(e_j = {-1}\), for \(j = 1, \ldots, m\).

Example: Word Arithmetic in a Fibonacci Group (ex-0eba9a)

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We illustrate the word operations by applying them to elements of the Fibonacci group \(F(2,5)\).

> FG<a,b,c,d,e> := FreeGroup(5);
> F := quo< FG | a*b=c, b*c=d, c*d=e, d*e=a, e*a=b>;
> G<a,b,c,d,e> := RWSGroup(F);
> a*b^-1;
e^-1
> a/b;
e^-1
> (c*d)^4;
a
> a^b, b^-1*a*b;
a a
> a^-2,
> Inverse(a)^2;
d d
> c^-1*d^-1*c*d eq (c,d);
true
> IsIdentity(a*b*c^-1);
true
> #(c*d);
1

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