This chapter describes functions to compute with elements of a
free noncommutative algebra.
The elements of the algebra are sums of rational multiples of words
in a free monoid.
These are called <E>monoid polynomials</E>,
and are stored as lists of pairs <C>[coefficient, word]</C>.
<Section><Heading>Construction of monoid polynomials</Heading>
<ManSection>
<Oper Name="MonoidPolyFromCoeffsWords"
Arg="coeffs, words" />
<Oper Name="MonoidPoly"
Arg="terms" />
<Oper Name="ZeroMonoidPoly"
Arg="F" />
<Description>
There are two ways to input a monoid polynomial:
by listing the coefficients and then the words;
or by listing the terms as a list of pairs <C>[coefficient, word]</C>.
If a word occurs more than once in the input list,
the coefficients will be added so that the terms of the monoid polynomial
recorded do not contain any duplicates.
The zero monoid polynomial is the polynomial with no terms.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> relq8 := RelatorsOfFpGroup( q8 );
[ f1^4, f2^4, f1*f2*f1*f2^-1, f1^2*f2^2 ]
gap> freeq8 := FreeGroupOfFpGroup( q8 );;
gap> gens := GeneratorsOfGroup( freeq8 );;
gap> famfree := ElementsFamily( FamilyObj( freeq8 ) );;
gap> famfree!.monoidPolyFam := MonoidPolyFam;;
gap> cg := [6,7];;
gap> pg := MonoidPolyFromCoeffsWords( cg, gens );;
gap> Print( pg, "\n" );
7*f2 + 6*f1
gap> cr := [3,4,-5,-2];;
gap> pr := MonoidPolyFromCoeffsWords( cr, relq8 );;
gap> Print( pr, "\n" );
4*f2^4 - 5*f1*f2*f1*f2^-1 - 2*f1^2*f2^2 + 3*f1^4
gap> Print( ZeroMonoidPoly( freeq8 ), "\n" );
zero monpoly
]]>
</Example>
</Section>
<Section><Heading>Components of a polynomial</Heading>
<ManSection>
<Attr Name="Terms"
Arg="poly" Label="for monoid polynomials" />
<Attr Name="Coeffs"
Arg="poly" />
<Attr Name="Words"
Arg="poly" />
<Attr Name="LeadTerm"
Arg="poly" Label="for monoid polynomials" />
<Attr Name="LeadCoeffMonoidPoly"
Arg="poly" />
<Description>
The function <C>Terms</C> returns the terms of a polynomial as a list of pairs
of the form <C>[word, coefficient]</C>.
The function <C>Coeffs</C> returns the coefficients of a polynomial as a list,
and the function <C>Words</C> returns the words of a polynomial as a list.
The function <C>LeadTerm</C> returns the term of the polynomial
whose word component is the largest with respect to
the length-lexicographical ordering.
The function <C>LeadCoeffMonoidPoly</C> returns the coefficient
of the leading term of a polynomial.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> Coeffs( pr );
[ 4, -5, -2, 3 ]
gap> Terms( pr );
[ [ 4, f2^4 ], [ -5, f1*f2*f1*f2^-1 ], [ -2, f1^2*f2^2 ], [ 3, f1^4 ] ]
gap> Words( pr );
[ f2^4, f1*f2*f1*f2^-1, f1^2*f2^2, f1^4 ]
gap> LeadTerm( pr );
[ 4, f2^4 ]
gap> LeadCoeffMonoidPoly( pr );
4
]]>
</Example>
<ManSection>
<Oper Name="Monic"
Arg="poly" />
<Description>
A monoid polynomial is called <E>monic</E> if the coefficient of
its leading polynomial is one.
The function <C>Monic</C> converts a polynomial into a monic polynomial
by dividing all the coefficients by the leading coefficient.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> mpr := Monic( pr );;
gap> Print( mpr, "\n" );
f2^4 - 5/4*f1*f2*f1*f2^-1 - 1/2*f1^2*f2^2 + 3/4*f1^4
]]>
</Example>
<ManSection>
<Oper Name="AddTermMonoidPoly"
Arg="poly, coeff, word" />
<Description>
The function <C>AddTermMonoidPoly</C> adds a new term,
given by its coeffiecient and word, to an existing polynomial.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> w := gens[1]^gens[2];
f2^-1*f1*f2
gap> cw := 3/4;;
gap> wpg := AddTermMonoidPoly( pg, cw, w );;
gap> Print( wpg, "\n" );
3/4*f2^-1*f1*f2 + 7*f2 + 6*f1
]]>
</Example>
</Section>
<Index>=,+,* for monoid polynomials</Index>
Tests for equality and arithmetic operations are performed in the usual way.
<P/>
The operation <C>poly1 = poly2</C> returns <K>true</K>
if the monoid polynomials have the same terms, and <K>false</K> otherwise.
Multiplication of a monoid polynomial (on the left or right)
by a coefficient;
the addition or subtraction of two monoid polynomials;
multiplication (on the right) of a monoid polynomial by a word;
and multiplication of two monoid polynomials; are all implemented.
<P/>
<Example>
<![CDATA[
gap> [ pg = pg, pg = pr ];
[ true, false ]
gap> prcw := pr * cw;;
gap> Print( prcw, "\n" );
3*f2^4 - 15/4*f1*f2*f1*f2^-1 - 3/2*f1^2*f2^2 + 9/4*f1^4
gap> cwpr := cw * pr;;
gap> Print( cwpr, "\n" );
3*f2^4 - 15/4*f1*f2*f1*f2^-1 - 3/2*f1^2*f2^2 + 9/4*f1^4
gap> [ pr = prcw, prcw = cwpr ];
[ false, true ]
gap> Print( pg + pr, "\n" );
4*f2^4 - 5*f1*f2*f1*f2^-1 - 2*f1^2*f2^2 + 3*f1^4 + 7*f2 + 6*f1
gap> Print( pg - pr, "\n" );
- 4*f2^4 + 5*f1*f2*f1*f2^-1 + 2*f1^2*f2^2 - 3*f1^4 + 7*f2 + 6*f1
gap> Print( pg * w, "\n" );
6*f1*f2^-1*f1*f2 + 7*f1*f2
gap> Print( pg * pr, "\n" );
28*f2^5 - 35*(f2*f1)^2*f2^-1 - 14*f2*f1^2*f2^2 + 21*f2*f1^4 + 24*f1*f2^4 -
30*f1^2*f2*f1*f2^-1 - 12*f1^3*f2^2 + 18*f1^5
]]>
</Example>
<ManSection>
<Meth Name="Length"
Arg="poly" Label="for monoid polynomials" />
<Description>
This function returns the number of distinct terms in the monoid polynomial.
<P/>
The boolean function <C>poly1 > poly2</C> returns <K>true</K>
if the first polynomial has more terms than the second.
If the polynomials are the same length it will compare their leading terms.
If the leading word of the first is lengthlexicographically greater
than the leading word of the second,
or if the words are equal but the coefficient of the first is greater
than the coefficient of the second then true is returned.
If the leading terms are equal then the next terms are compared
in the same way.
If all terms are the same then <K>false</K> is returned.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> Length( pr );
4
gap> [ pr > 3*pr, pr > pg ];
[ false, true ]
]]>
</Example>
</Section>
<Section><Heading>Reduction of a Monoid Polynomial</Heading>
<ManSection Label="reduce-monoid-poly">
<Oper Name="ReduceMonoidPoly"
Arg="poly, rules" />
<Description>
Recall that the words of a monoid polynomial are elements of a free monoid.
Given a rewrite system (set of rules) on the free monoid
the words can be reduced.
This allows us to simulate calculation in monoid rings
where the monoid is given by a complete presentation.
This function reduces the words of the polynomial
(elements of the free monoid) with respect to the complete rewrite system.
The words of the reduced polynomial are normal forms for the elements
of the monoid presented by that rewite system.
The list of rules <C>r2</C> is displayed in section 2.3.3.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> M := genfmq8;;
gap> mp1 := MonoidPolyFromCoeffsWords( [9,-7,5],
> [ M[1]*M[3], M[2]^3, M[4]*M[3]*M[2] ] );;
gap> PrintUsingLabels( mp1, genfmq8, q8labs );
5*B*A*b + -7*b^3 + 9*a*A
gap> rmp1 := ReduceMonoidPoly( mp1, r2 );;
gap> PrintUsingLabels( rmp1, genfmq8, q8labs );
-7*B + 5*a + 9*id
]]>
</Example>
</Section>
</Chapter>
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