Difference between revisions of "ApCoCoA-1:FreeAbelian groups"
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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. A representation is given by the | 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>
