Arithmetic with Words

[Next][Prev] [Right] [Left] [Up] [Index] [Root]

Arithmetic with Words

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

the product w = u * v is formed as a product in the appropriate free monoid.
w is reduced using the reduction machine associated with M.

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

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.
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 : MonRWSElt, MonRWSElt -> MonRWSElt

Product of the words w and v.

u ^ n : MonRWSElt, RngIntElt -> MonRWSElt

The n-th power of the word w, where n is a positive or zero integer.

u eq v : MonRWSElt, MonRWSElt -> BoolElt

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

u ne v : MonRWSElt, MonRWSElt -> BoolElt

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

IsId(w) : MonRWSElt -> BoolElt

IsIdentity(w) : MonRWSElt -> BoolElt

True if the word w is the identity word.

# u : MonRWSElt -> RngIntElt

The length of the word w.

ElementToSequence(u) : MonRWSElt -> [ RngIntElt ]

Eltseq(u) : MonRWSElt -> [ RngIntElt ]

The sequence Q obtained by decomposing the element u of a rewrite monoid into its constituent generators. Suppose u is a word in the rewrite monoid M. If u = M.i_1 ... M.i_m, then Q[j] = i_j, for j = 1, ..., m.

Example `MonRWS_Arithmetic (H18E3)`

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

> FM<a,A,b,B,c,C,d,D,e,E> := FreeMonoid(10);
> Q := quo< FM | a*A=1, A*a=1, b*B=1, B*b=1, c*C=1, 
>         C*c=1, d*D=1, D*d=1, e*E=1, E*e=1,
>         a*b=c, b*c=d, c*d=e, d*e=a, e*a=b>;
> M<a,A,b,B,c,C,d,D,e,E> := RWSMonoid(Q);
> print a*b*c*d;
E
> print (c*d)^4 eq a;
true
> print IsIdentity(a*A);
true
> print #(d*a*(B*b)^10);
1

[Next][Prev] [Right] [Left] [Up] [Index] [Root]