Difference between revisions of "ApCoCoA-1:NC.KernelOfHomomorphism"

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.KernelOfHomomorphism</title>
 
<title>NC.KernelOfHomomorphism</title>
 
<short_description>
 
<short_description>
(Partial) (two-sided) leading-term ideal of the kernel of a <tt>K</tt>-algebra homomorphism.
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The kernel of an algebra homomorphism.
 
</short_description>
 
</short_description>
<syntax>
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<syntax></syntax>
NC.KernelOfHomomorphism(X1:STRING, X2:STRING, Images:LIST):LIST
 
NC.KernelOfHomomorphism(X1:STRING, X2:STRING, Images:LIST, DegreeBound:INT, LoopBound:INT, Flag:INT):LIST
 
</syntax>
 
 
<description>
 
<description>
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
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<em>Proposition (Kernel of an Algebra Homomorphism):</em> Let <tt>I</tt> be a two-sided ideal in the free monoid ring <tt>K&lt;x[1],...,x[n]&gt;</tt>, and let <tt>J</tt> be a two-sided ideal in the free monoid ring <tt>K&lt;y[1],...,y[m]&gt;</tt>. Moreover, let <tt>g_1,...,g_m</tt> be polynomials in <tt>K&lt;x[1],...,x[n]&gt;</tt>, and let <tt>phi: K&lt;y[1],...,y[m]&gt;/J--&gt;K&lt;x[1],...,x[n]&gt;/I</tt> be a homomorphism of <tt>K</tt>-algebras defined by <tt>phi(y[i]+J)=g_i+I</tt> for <tt>i=1,...,m</tt>. We form the free monoid ring <tt>K&lt;x[1],...,x[n],y[1],...,y[m]&gt;</tt>, and let <tt>D</tt> be the diagonal ideal generated by the set <tt>{y[1]-g_1,...,y[m]-g_m}</tt>. Then we have <tt>ker(phi)=((D+J) intersets K&lt;y[1],...,y[m]&gt;)+I</tt>.
<par/>
 
Please set ring environment coefficient field <tt>K</tt> and alphabet (or indeterminates) <tt>X</tt> through the functions <ref>NC.SetFp</ref>(Prime) and <ref>NC.SetX</ref>(X), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
 
<itemize>
 
<item>@param <em>X1:</em> a finite set of letters. It is of STRING type. Note that every letter in X1 MUST appear only once.The order of letters in X1 is very important, since X1 as well as <tt>Images</tt> are used to define a K-algebra homomorphism.</item>
 
<item>@param <em>X2:</em> another finite set of letters. It is of STRING type. Note that X1 and X2 are disjoint.</item>
 
<item>@param <em>Images:</em> a LIST of polynomials in <tt>K&lt;X2&gt;</tt>. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST []. Note that the order of polynomials in Images is very important, since Images as well as X1 are used to defined a K-algebra homomorphism. For example, X1:=<quotes>abc</quotes>; F1 := [[1,<quotes>x</quotes>], [1,<quotes>y</quotes>]]; F2 := [[1,<quotes>xx</quotes>],[1,<quotes>xy</quotes>]]; F3 := [[1,<quotes>yy</quotes>],[1,<quotes>yx</quotes>]]; Images := [F1, F2, F3]; these together define a K-algebra homomorphism by mapping a to F1, b to F2 and c to F3.</item>
 
</itemize>
 
Since the algorithm used in this function is based on Groebner basis computation, we refer users to <ref>NC.GB</ref> or <ref>NC.ReducedGB</ref> for information about the following optional parameters:
 
<itemize>
 
<item>@param <em>DegreeBound</em></item>
 
<item>@param <em>LoopBound</em></item>
 
<item>@param <em>Flag</em></item>
 
<item>@return: a LIST of polynomials, which is a Groebner basis of the kernel of a <tt>K</tt>-algebra homomorphism if a finite Groebner basis exists, and is a partial Groebner basis otherwise.</item>
 
</itemize>
 
 
<example>
 
<example>
X1 := <quotes>abc</quotes>;  
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Use QQ[x[1..2],y[1..2],a,b];
X2 := <quotes>xy</quotes>;  
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NC.SetOrdering("ELIM"); -- we want to eliminate x[1..2],y[1..2]
F1 := [[1,<quotes>x</quotes>], [1,<quotes>y</quotes>]];  
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-- Group ring Q&lt;a,b&gt;/&lt;F11,F12,F13&gt; with the triangle group of order 576
F2 := [[1,<quotes>xx</quotes>],[1,<quotes>xy</quotes>]];  
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F11:=[[a^2], [-1]];
F3 := [[1,<quotes>yy</quotes>],[1,<quotes>yx</quotes>]];  
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F12:=[[b^3], [-1]];
Images :=[F1, F2, F3]; -- a |-> F1; b |-> F2; c |-> F3
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F13:=[[a,b,a,b,a,b^2,a,b^2,a,b,a,b,a,b^2,a,b^2], [-1]];
NC.KernelOfHomomorphism(X1, X2, Images);
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-- Group ring Q&lt;x[1],y[1],x[2],y[2]&gt;/&lt;F21,F22,F23,F24,F25,F26&gt; with the fundamental group of 3-manifold of order 72
[[[1, <quotes>ab</quotes>], [-1, <quotes>ba</quotes>], [1, <quotes>ac</quotes>], [-1, <quotes>ca</quotes>]], [[1, <quotes>aa</quotes>], [-1, <quotes>b</quotes>], [-1, <quotes>c</quotes>]]]
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F21:=[[x[1]^3],[y[1]^3]]; --x[1]^3=y[1]^3
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F22:=[[x[1]^3],[-y[1],x[2],y[1],x[2]]]; --x[1]^3p=(y[1]x[1]^-1)^2
 +
F23:=[[x[2],x[1]],[-1]];
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F24:=[[x[1],x[2]],[-x[2],x[1]]];
 +
F25:=[[y[2],y[1]],[-1]];
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F26:=[[y[1],y[2]],[-y[2],y[1]]];
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-- Q-group algebra homomorphism phi: Q&lt;a,b&gt;/&lt;F11,F12,F13&gt; --&gt;Q&lt;x[1],y[1],x[2],y[2]&gt;/&lt;F21,F22,F23,F24,F25,F26&gt;  
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-- defined by mapping a to x[1] and b to 0
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D1:=[[a],[-x[1]]]; -- map a to x[1]
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D2:=[[b]]; -- map b to 0
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G:=[D1,D2,F21,F22,F23,F24,F25,F26];
 +
Gb:=NC.GB(G);
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KGb:=NC.FindPolys(Gb,[a,b]);
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Concat(KGb,[F11,F12,F13]); -- a generating sy[1]stem of the ker(phi)
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[[[b]], [[a^18], [-1]], [[a^2], [-1]], [[b^3], [-1]],  
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[[a, b, a, b, a, b^2, a, b^2, a, b, a, b, a, b^2, a, b^2], [-1]]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 +
<em>Corollary (Minimal Polynomial):</em> Let <tt>phi: K[y]--&gt;K&lt;x[1],...,x[n]&gt;/I</tt> be a <tt>K</tt>-algebra homomorphism given by <tt>phi(y)=g+I</tt>. Then <tt>g+I</tt> is algebraic over <tt>K</tt> if and only if <tt>ker(phi)</tt> is not zero. Moreover, if <tt>g+I</tt> is algebraic over <tt>K</tt>, then the unique monic generating polynomial of the ideal <tt>ker(phi)</tt> is the minimal polynomial of <tt>g+I</tt> over <tt>K</tt>.
 +
<example>
 +
-- The following example shows that a is transcendental over Q in the algebra Q&lt;a,b&gt;/&lt;b^2-1,(ab)^2-1&gt;,
 +
-- hence the group &lt;a,b: b^2=(ab)^2=1&gt; is infinite.
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Use QQ[a,b,y];
 +
NC.SetOrdering("ELIM"); -- we want to eliminate a and b
 +
F1:=[[b^2],[-1]]; -- b^2-1
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F2:=[[a,b,a,b],[-1]]; -- (ab)^2-1
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-- Construct a Q-algebra homomorphism phi: Q[y]--&gt; Q&lt;a,b&gt;/&lt;b^2-1,(ab)^2-1&gt;
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-- defined by mapping y to a
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D:=[[y],[-a]]; -- y-a
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G:=[D,F1,F2];
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Gb:=NC.GB(G);
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Gb;
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NC.FindPolys(Gb,[y]);
 +
 +
[[[a], [-y]], [[b^2], [-1]], [[y, b, y], [-b]]]
 +
-------------------------------
 +
[ ]
 +
-------------------------------
 +
Use QQ[a,b,y];
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NC.SetOrdering("ELIM"); -- we want to eliminate a and b
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G1:=[[a^3],[-1]]; -- a^3-1
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G2:=[[b^2],[-1]]; -- b^2-1
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G3:=[[a,b,a,b],[-1]]; -- (ab)^2-1
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-- Construct a Q-algebra homomorphism phi: Q[y]--&gt; Q&lt;a,b&gt;/&lt;a^3-1,b^2-1,(ab)^2-1&gt;
 +
-- defined by mapping y to bab
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D:=[[y],[-b,a,b]]; -- y-bab
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G:=[D,G1,G2,G3];
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Gb:=NC.GB(G);
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NC.FindPolys(Gb,[y]);
 +
 +
[[[y^3], [-1]]]
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-------------------------------
 +
</example>
 +
<par/>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NC.Add</see>
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<see>ApCoCoA-1:Use|Use</see>
<see>NC.Deg</see>
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<see>ApCoCoA-1:NC.FindPolys|NC.FindPolys</see>
<see>NC.FindPolynomials</see>
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<see>ApCoCoA-1:NC.GB|NC.GB</see>
<see>NC.GB</see>
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<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
<see>NC.HF</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
<see>NC.Intersection</see>
 
<see>NC.IsGB</see>
 
<see>NC.KernelOfHomomorphism</see>
 
<see>NC.LC</see>
 
<see>NC.LT</see>
 
<see>NC.LTIdeal</see>
 
<see>NC.MinimalPolynomial</see>
 
<see>NC.Multiply</see>
 
<see>NC.NR</see>
 
<see>NC.ReducedGB</see>
 
<see>NC.SetFp</see>
 
<see>NC.SetOrdering</see>
 
<see>NC.SetRelations</see>
 
<see>NC.SetRules</see>
 
<see>NC.SetX</see>
 
<see>NC.Subtract</see>
 
<see>NC.UnsetFp</see>
 
<see>NC.UnsetOrdering</see>
 
<see>NC.UnsetRelations</see>
 
<see>NC.UnsetRules</see>
 
<see>NC.UnsetX</see>
 
<see>Introduction to CoCoAServer</see>
 
 
</seealso>
 
</seealso>
 
<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
 
<type>ideal</type>
 
<type>ideal</type>
 +
<type>groebner</type>
 
<type>non_commutative</type>
 
<type>non_commutative</type>
 
</types>
 
</types>
<key>gbmr.KernelOfHomomorphism</key>
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<key>ncpoly.KernelOfHomomorphism</key>
 
<key>NC.KernelOfHomomorphism</key>
 
<key>NC.KernelOfHomomorphism</key>
 
<key>KernelOfHomomorphism</key>
 
<key>KernelOfHomomorphism</key>
<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:34, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.KernelOfHomomorphism

The kernel of an algebra homomorphism.

Syntax

Description

Proposition (Kernel of an Algebra Homomorphism): Let I be a two-sided ideal in the free monoid ring K<x[1],...,x[n]>, and let J be a two-sided ideal in the free monoid ring K<y[1],...,y[m]>. Moreover, let g_1,...,g_m be polynomials in K<x[1],...,x[n]>, and let phi: K<y[1],...,y[m]>/J-->K<x[1],...,x[n]>/I be a homomorphism of K-algebras defined by phi(y[i]+J)=g_i+I for i=1,...,m. We form the free monoid ring K<x[1],...,x[n],y[1],...,y[m]>, and let D be the diagonal ideal generated by the set {y[1]-g_1,...,y[m]-g_m}. Then we have ker(phi)=((D+J) intersets K<y[1],...,y[m]>)+I.

Example

Use QQ[x[1..2],y[1..2],a,b];
NC.SetOrdering("ELIM"); -- we want to eliminate x[1..2],y[1..2]
-- Group ring Q&lt;a,b&gt;/&lt;F11,F12,F13&gt; with the triangle group of order 576
F11:=[[a^2], [-1]];
F12:=[[b^3], [-1]];
F13:=[[a,b,a,b,a,b^2,a,b^2,a,b,a,b,a,b^2,a,b^2], [-1]];
-- Group ring Q&lt;x[1],y[1],x[2],y[2]&gt;/&lt;F21,F22,F23,F24,F25,F26&gt; with the fundamental group of 3-manifold of order 72
F21:=[[x[1]^3],[y[1]^3]]; --x[1]^3=y[1]^3
F22:=[[x[1]^3],[-y[1],x[2],y[1],x[2]]]; --x[1]^3p=(y[1]x[1]^-1)^2
F23:=[[x[2],x[1]],[-1]];
F24:=[[x[1],x[2]],[-x[2],x[1]]];
F25:=[[y[2],y[1]],[-1]];
F26:=[[y[1],y[2]],[-y[2],y[1]]];
-- Q-group algebra homomorphism phi: Q&lt;a,b&gt;/&lt;F11,F12,F13&gt; --&gt;Q&lt;x[1],y[1],x[2],y[2]&gt;/&lt;F21,F22,F23,F24,F25,F26&gt; 
-- defined by mapping a to x[1] and b to 0
D1:=[[a],[-x[1]]]; -- map a to x[1]
D2:=[[b]]; -- map b to 0
G:=[D1,D2,F21,F22,F23,F24,F25,F26];
Gb:=NC.GB(G);
KGb:=NC.FindPolys(Gb,[a,b]);
Concat(KGb,[F11,F12,F13]); -- a generating sy[1]stem of the ker(phi)

[[[b]], [[a^18], [-1]], [[a^2], [-1]], [[b^3], [-1]], 
[[a, b, a, b, a, b^2, a, b^2, a, b, a, b, a, b^2, a, b^2], [-1]]]
-------------------------------

Corollary (Minimal Polynomial): Let phi: K[y]-->K<x[1],...,x[n]>/I be a K-algebra homomorphism given by phi(y)=g+I. Then g+I is algebraic over K if and only if ker(phi) is not zero. Moreover, if g+I is algebraic over K, then the unique monic generating polynomial of the ideal ker(phi) is the minimal polynomial of g+I over K.

Example

-- The following example shows that a is transcendental over Q in the algebra Q&lt;a,b&gt;/&lt;b^2-1,(ab)^2-1&gt;,
-- hence the group &lt;a,b: b^2=(ab)^2=1&gt; is infinite.
Use QQ[a,b,y];
NC.SetOrdering("ELIM"); -- we want to eliminate a and b
F1:=[[b^2],[-1]]; -- b^2-1
F2:=[[a,b,a,b],[-1]]; -- (ab)^2-1
-- Construct a Q-algebra homomorphism phi: Q[y]--&gt; Q&lt;a,b&gt;/&lt;b^2-1,(ab)^2-1&gt;
-- defined by mapping y to a
D:=[[y],[-a]]; -- y-a
G:=[D,F1,F2];
Gb:=NC.GB(G);
Gb;
NC.FindPolys(Gb,[y]);

[[[a], [-y]], [[b^2], [-1]], [[y, b, y], [-b]]]
-------------------------------
[ ]
-------------------------------
Use QQ[a,b,y];
NC.SetOrdering("ELIM"); -- we want to eliminate a and b
G1:=[[a^3],[-1]]; -- a^3-1
G2:=[[b^2],[-1]]; -- b^2-1
G3:=[[a,b,a,b],[-1]]; -- (ab)^2-1
-- Construct a Q-algebra homomorphism phi: Q[y]--&gt; Q&lt;a,b&gt;/&lt;a^3-1,b^2-1,(ab)^2-1&gt;
-- defined by mapping y to bab
D:=[[y],[-b,a,b]]; -- y-bab
G:=[D,G1,G2,G3];
Gb:=NC.GB(G);
NC.FindPolys(Gb,[y]);

[[[y^3], [-1]]]
-------------------------------


See also

Use

NC.FindPolys

NC.GB

NC.SetOrdering

Introduction to CoCoAServer