ApCoCoA-1:FreeAbelian groups: Difference between revisions
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New page: === <div id="FreeAbelian_groups">Free abelian groups</div> === ==== Description ==== Every element in a free abelian group can be written in o... |
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=== <div id="FreeAbelian_groups">[[:ApCoCoA:Symbolic data#Free_abelian_group|Free | === <div id="FreeAbelian_groups">[[:ApCoCoA:Symbolic data#Free_abelian_group|Free Abelian Groups]]</div> === | ||
==== Description ==== | ==== Description ==== | ||
Every element in a free abelian group can be written in only way as a finite linear combination. | Every element in a free abelian group can be written in only way as a finite linear combination. A representation is given by the | ||
following: | following: | ||
Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j> | Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j> | ||
==== Reference ==== | |||
Phillip A. Griffith, Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press, 1970. | |||
==== Computation ==== | ==== Computation ==== | ||
/*Use the ApCoCoA package ncpoly.*/ | |||
// Number of free abelian group | |||
MEMORY.N:=3; | |||
Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | |||
NC.SetOrdering("LLEX"); | |||
Define CreateRelationsFreeAbelian() | |||
Relations:=[]; | |||
// add the relations of the inverse elements | |||
For Index1 := 1 To MEMORY.N Do | |||
Append(Relations,[[x[Index1],y[Index1]],[1]]); | |||
Append(Relations,[[y[Index1],x[Index1]],[1]]); | |||
EndFor; | |||
// add the relations [x_{i},x_{j}]=1 | |||
For Index1 := 1 To MEMORY.N Do | |||
For Index2 := 1 To MEMORY.N Do | |||
Append(Relations,[[x[Index1],x[Index2],y[Index1],y[Index2]],[1]]); | |||
EndFor; | |||
EndFor; | |||
Return Relations; | |||
EndDefine; | |||
Relations:=CreateRelationsFreeAbelian(); | |||
Relations; | |||
Gb:=NC.GB(Relations,31,1,100,1000); | |||
Gb; | |||
====Example in Symbolic Data Format==== | |||
<FREEALGEBRA createdAt="2014-06-05" createdBy="strohmeier"> | |||
<vars>x1,x2,x3,y1,y2,y3</vars> | |||
<uptoDeg>11</uptoDeg> | |||
<basis> | |||
<ncpoly>x1*y1-1</ncpoly> | |||
<ncpoly>y1*x1-1</ncpoly> | |||
<ncpoly>x2*y2-1</ncpoly> | |||
<ncpoly>y2*x2-1</ncpoly> | |||
<ncpoly>x3*y3-1</ncpoly> | |||
<ncpoly>y3*x3-1</ncpoly> | |||
<ncpoly>x1*x1*y1*y1-1</ncpoly> | |||
<ncpoly>x1*x2*y1*y2-1</ncpoly> | |||
<ncpoly>x1*x3*y1*y3-1</ncpoly> | |||
<ncpoly>x2*x1*y2*y1-1</ncpoly> | |||
<ncpoly>x2*x2*y2*y2-1</ncpoly> | |||
<ncpoly>x2*x3*y2*y3-1</ncpoly> | |||
<ncpoly>x3*x1*y3*y1-1</ncpoly> | |||
<ncpoly>x3*x2*y3*y2-1</ncpoly> | |||
<ncpoly>x3*x3*y3*y3-1</ncpoly> | |||
</basis> | |||
<Comment>The partial LLex Gb has 180 elements</Comment> | |||
<Comment>Free Abelian Group_3</Comment> | |||
</FREEALGEBRA> |
Latest revision as of 10:49, 6 May 2014
Description
Every element in a free abelian group can be written in only way as a finite linear combination. A representation is given by the following:
Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j>
Reference
Phillip A. Griffith, Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press, 1970.
Computation
/*Use the ApCoCoA package ncpoly.*/ // Number of free abelian group MEMORY.N:=3; Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; NC.SetOrdering("LLEX"); Define CreateRelationsFreeAbelian() Relations:=[]; // add the relations of the inverse elements For Index1 := 1 To MEMORY.N Do Append(Relations,[[x[Index1],y[Index1]],[1]]); Append(Relations,[[y[Index1],x[Index1]],[1]]); EndFor; // add the relations [x_{i},x_{j}]=1 For Index1 := 1 To MEMORY.N Do For Index2 := 1 To MEMORY.N Do Append(Relations,[[x[Index1],x[Index2],y[Index1],y[Index2]],[1]]); EndFor; EndFor; Return Relations; EndDefine; Relations:=CreateRelationsFreeAbelian(); Relations; Gb:=NC.GB(Relations,31,1,100,1000); Gb;
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-06-05" createdBy="strohmeier"> <vars>x1,x2,x3,y1,y2,y3</vars> <uptoDeg>11</uptoDeg> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x1*x1*y1*y1-1</ncpoly> <ncpoly>x1*x2*y1*y2-1</ncpoly> <ncpoly>x1*x3*y1*y3-1</ncpoly> <ncpoly>x2*x1*y2*y1-1</ncpoly> <ncpoly>x2*x2*y2*y2-1</ncpoly> <ncpoly>x2*x3*y2*y3-1</ncpoly> <ncpoly>x3*x1*y3*y1-1</ncpoly> <ncpoly>x3*x2*y3*y2-1</ncpoly> <ncpoly>x3*x3*y3*y3-1</ncpoly> </basis> <Comment>The partial LLex Gb has 180 elements</Comment> <Comment>Free Abelian Group_3</Comment> </FREEALGEBRA>