<Heading>Functions for Noncommutative Monomials</Heading>
A monomial, such as <M>ab^2a</M> is represented in <Package>GBNP</Package>
as the list <M>[1,2,2,1]</M>.
Polynomials have a more complicated structure, for example
<M>6ab^2a - 7ab + 8ba</M> is represented in <Package>GBNP</Package> by
<M>[ [ [1,2,2,1], [1,2], [2,1] ], [6,-7,8] ]</M>, which is a list
of monomials followed by the corresponding list of coefficients.
Polynomials are dealt with in the following chapter.
<P/>
As shown in Section <Ref Sect="sec-nps"/>,
<Package>GBNP</Package> has functions <C>PrintNP</C> and <C>PrintNPList</C>
to print a polynomial and a list of polynomials.
Here we provide equivalent functions for monomials.
<Section Label="sec-basic">
<Heading>Basic functions for monomials</Heading>
<Subsection Label="sub-inbuilt-alg">
<Heading>Predefined algebras</Heading>
For convenience of use in examples, three algebras over the rationals,
<C>AlbebraIBNP</C> and <C>AlgebrakIBNP</C> with <M>k \in [2,3,4]</M>,
are predefined in this package.
<Index>AlgebraIBNP</Index>
<Example>
<![CDATA[
gap> GeneratorsOfAlgebra( AlgebraIBNP );
[ (1)*<identity ...>, (1)*a, (1)*b ]
gap> Algebra2IBNP = AlgebraIBNP;
true
gap> A3 := Algebra3IBNP;
<algebra-with-one over Rationals, with 3 generators>
]]>
</Example>
</Subsection>
<ManSection>
<Oper Name="PrintNM"
Arg="monomial" />
<Oper Name="PrintNMList"
Arg="list" />
<Description>
Recall, from <Package>GBNP</Package>, that the actual letters printed
are controlled by the operation <C>GBNP.ConfigPrint</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> GBNP.ConfigPrint( "a", "b", "c" );
gap> mon := [2,1,1,1,3,3,1];;
gap> PrintNM( mon );
ba^3c^2a
gap> L := [ [1,2,2], [3,1,2], [3,3,3], [2], [ ] ];;
gap> PrintNMList( L );
ab^2
cab
c^3
b
1
]]>
</Example>
<ManSection>
<Oper Name="NM2GM"
Arg="monomial algebra" />
<Oper Name="NM2GMList"
Arg="list algebra" />
<Description>
Recall, from <Package>GBNP</Package>, that the functions <C>NP2GP</C>
and <C>NP2GPList</C> convert a polynomial (or list of polynomials)
in NP-format to an element of the algebra.
This package provides additional functions <C>NM2GM</C> and <C>NM2GMList</C>
which do the equivalent conversion for monomials.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> m := NM2GM( mon, A3 );
(1)*b*a^3*c^2*a
gap> NM2GMList( [ mon, Reversed(mon), Concatenation(mon,mon) ], A3 );
[ (1)*b*a^3*c^2*a, (1)*a*c^2*a^3*b, (1)*(b*a^3*c^2*a)^2 ]
]]>
</Example>
<ManSection>
<Oper Name="GM2NM"
Arg="monomial" />
<Oper Name="GM2NMList"
Arg="list" />
<Description>
Recall, from <Package>GBNP</Package>, that the functions <C>GP2NP</C>
and <C>GP2NPList</C> convert a polynomial (or list of polynomials)
to the equivalent NP-format.
This package provides additional functions <C>GM2NM</C> and <C>GM2NMList</C>
which do the equivalent conversion for monomials.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> a:=A3.1;; b:=A3.2;; c:=A3.3;;
gap> p := (a*b*c)^2;;
gap> GM2NM(p);
[ 1, 2, 3, 1, 2, 3 ]
gap> GM2NMList( [ p, p^2, a^3, b^4, c^5 ] );
[ [ 1, 2, 3, 1, 2, 3 ], [ 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3 ], [ 1, 1, 1 ],
[ 2, 2, 2, 2 ], [ 3, 3, 3, 3, 3 ] ]
]]>
</Example>
<ManSection>
<Oper Name="PrefixNM"
Arg="monomial posint" />
<Oper Name="SubwordNM"
Arg="monomial posint posint" />
<Oper Name="SuffixNM"
Arg="monomial posint" />
<Description>
These are the three operations which pick a sublist from a monomial list.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> mon := [2,1,1,1,3,3,1];;
gap> PrefixNM( mon, 3 );
[ 2, 1, 1 ]
gap> SubwordNM( mon, 3, 6 );
[ 1, 1, 3, 3 ]
gap> SuffixNM( mon, 3 );
[ 3, 3, 1 ]
]]>
</Example>
<ManSection>
<Oper Name="SuffixPrefixPosNM"
Arg="monomial monomial posint posint" />
<Description>
The operation <C>SuffixPrefixPosNM( left, right, start, limit)</C>
looks for overlaps of type <E>suffix of left = prefix of right</E>.
The size of the smallest such overlap is returned.
The overlaps which are considered are controlled by the third and fourth
arguments.
We commence by looking at the overlap of size <E>start</E>
and go no further than the overlap of size <E>limit</E>.
When no overlap exists, <M>0</M> is returned.
To test all possibilities, <E>start</E> should be <M>1</M>
and <E>limit</E> should be <M>min(|left|,|right|)-1</M>.
It is the user's responsibility to make sure that these bounds
are correct - no checks are made.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> m1 := [2,1,1,1,2,2,1,1];; ## m1 = ba^3b^2a^2
gap> m2 := [1,1,2,2,1,1];; ## m2 = a^2b^2a^2
gap> SuffixPrefixPosNM( m1, m2, 1, 5 ); ## overlap is a
1
gap> SuffixPrefixPosNM( m1, m2, 2, 5 ); ## overlap is a^2
2
gap> SuffixPrefixPosNM( m1, m2, 3, 5 ); ## no longer an overlap
0
gap> SuffixPrefixPosNM( m2, m1, 1, 5 ); ## overlap is ba^2
3
]]>
</Example>
<ManSection>
<Oper Name="SubwordPosNM"
Arg="monomial monomial posint" />
<Oper Name="IsSubwordNM"
Arg="monomial monomial" />
<Description>
The operation <C>SubwordPosNM( small, large, start );</C> answers the
question for monomials <E>Is small a subword of large?</E>.
The value returned is the start position in <E>large</E>
of the first subword found.
When no subword is found, <M>0</M> is returned.
The search commences at position <E>start</E> in <E>large</E> so,
to test all possibilities, the third argument should be <M>1</M>.
<P/>
To just ask whether or not <E>small</E> is a subword
of <E>large</E>, use <C>IsSubwordNM( small, large);</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> m3 := [ 1, 1, 2 ];; ## m3 = a^2b
gap> SubwordPosNM( m3, m1, 1 );
3 ## m1 = ba(a^b)ba^2
gap> SubwordPosNM( m3, m2, 1 );
1 ## m2 = (a^2b)ba^2
gap> SubwordPosNM( m3, m2, 2 );
0
gap> IsSubwordNM( [ 2, 1, 2 ], m1 );
false
]]>
</Example>
<ManSection>
<Oper Name="LeadVarNM"
Arg="monomial" />
<Oper Name="LeadExpNM"
Arg="monomial" />
<Oper Name="TailNM"
Arg="monomial" />
<Description>
Given the word <M>w = b^4a^3c^2</M>, represented by
<M>[2,2,2,2,1,1,1,3,3]</M>, the <E>lead variable</E> is <M>b</M>
or <M>2</M>, and the <E>lead exponent</E> is <M>4</M>.
Removing <M>b^4</M> from <M>w</M> leaves the <E>tail</E> <M>a^3c^2</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> mon4 := [2,2,2,2,1,1,1,3,3];;
gap> LeadVarNM( mon4 );
2
gap> LeadExpNM( mon4 );
4
gap> TailNM( mon4 );
[ 1, 1, 1, 3, 3 ]
]]>
</Example>
<ManSection>
<Oper Name="DivNM"
Arg="monomial monomial" />
<Description>
The operation <C>DivNM(large, small);</C> for two monomials
returns all the ways that <E>small</E> divides <E>large</E>
in the form of a list of pairs of monomials <E>[left,right]</E>
so that <E>large = left*small*right</E>.
In the example we search for subwords <M>ab</M> of <M>m = abcababc</M>,
returning <M>[ [abcab,c], [abc,abc], [1,cababc] ]</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> GBNP.ConfigPrint( "a", "b", "c" );
gap> m := [ 1, 2, 3, 1, 2, 1, 2, 3 ];;
gap> d := [ 1, 2 ];;
gap> PrintNMList( [ m, d ] );
abcababc
ab
gap> divs := DivNM( m, d );
[ [ [ 1, 2, 3, 1, 2 ], [ 3 ] ], [ [ 1, 2, 3 ], [ 1, 2, 3 ] ],
[ [ ], [ 3, 1, 2, 1, 2, 3 ] ] ]
gap> PrintNMList( divs[1] );
abcab
c
]]>
</Example>
</Section>
</Chapter>
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