Difference between revisions of "ApCoCoA-1:NC.Intersection"
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<title>NC.Intersection</title> | <title>NC.Intersection</title> | ||
<short_description> | <short_description> | ||
| − | Intersection of two finitely generated two-sided ideals in a | + | Intersection of two finitely generated two-sided ideals in a non-commutative polynomial ring. |
</short_description> | </short_description> | ||
<description> | <description> | ||
| − | <em>Propostion (Intersection of Two Ideals):</em> Let <tt>G_I</tt> and <tt>G_J</tt> be two sets of non-zero polynomials in the | + | <em>Propostion (Intersection of Two Ideals):</em> Let <tt>G_I</tt> and <tt>G_J</tt> be two sets of non-zero polynomials in the non-commutative polynomial ring <tt>K<x[1],...,x[n]></tt>, and let <tt>I</tt> and <tt>J</tt> be two ideals generated by <tt>G_I</tt> and <tt>G_J</tt>, respectively. We choose a new indeterminate <tt>y</tt>, and form the free monoid ring <tt>K<y,x[1],...,x[n]></tt>. Furthermore, let <tt>N</tt> be the ideal generated by the union of <tt>{yf: f in G_I}</tt> and <tt>{(1-y)g: g in G_J}</tt>, and let <tt>C</tt> be the ideal generated by the set <tt>{yx[1]-x[1]y,...,yx[n]-x[n]y}</tt> of commutators. Then we have the intersection of <tt>I</tt> and <tt>J</tt> is equal to the intersection of <tt>N+C</tt> and <tt>K<x[1],...,x[n]></tt>. |
<example> | <example> | ||
-- Let I be the ideal generated by G_I={xy+z,yz+x}, and J be the ideal generated by G_J={yz+x, zx+y}. | -- Let I be the ideal generated by G_I={xy+z,yz+x}, and J be the ideal generated by G_J={yz+x, zx+y}. | ||
-- We compute the intersection of I and J as follows. | -- We compute the intersection of I and J as follows. | ||
| − | + | Use QQ[t,x,y,z]; | |
NC.SetOrdering("ELIM"); -- Choose an elimination word ordering for t | NC.SetOrdering("ELIM"); -- Choose an elimination word ordering for t | ||
| − | F1 := [[ | + | F1 := [[x,y], [z]]; -- xy+z |
| − | F2 := [[ | + | F2 := [[y,z], [x]]; -- yz+x |
| − | G1 := [[ | + | G1 := [[y,z], [x]]; -- yz+x |
| − | G2 := [[ | + | G2 := [[z,x], [y]]; -- zx+y |
| − | N:=[NC. | + | N:=[NC.Mul([[t]],F1), NC.Mul([[t]],F2)]; -- t*F1, t*F2 |
| − | N:=Concat(N,[NC. | + | N:=Concat(N,[NC.Mul([[1],[-t]],G1), NC.Mul([[1],[-t]],G2)]); -- (1-t)*G1, (1-t)*G2 |
| − | C:=[[[ | + | C:=[[[t,x],[-x,t]], [[t,y],[-y,t]], [[t,z],[-z,t]]]; -- set of commutators |
G:=Concat(N,C); | G:=Concat(N,C); | ||
| − | Gb:=NC.GB(G,20,50 | + | Gb:=NC.GB(G,31,1,20,50); |
-- Done. | -- Done. | ||
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The following information printed by the ApCoCoAServer shows that Gb it is a partial Groebner basis. | The following information printed by the ApCoCoAServer shows that Gb it is a partial Groebner basis. | ||
the number of unselected generators: 0 | the number of unselected generators: 0 | ||
| − | the number of unselected | + | the number of unselected obstructions: 81 |
the procedure is interrupted by loop bound! | the procedure is interrupted by loop bound! | ||
| − | the total number of | + | the total number of obstructions: 293 |
| − | the number of selected | + | the number of selected obstructions: 43 |
| − | the number of | + | the number of obstructions detected by Criterion M: 128 |
| − | the number of | + | the number of obstructions detected by Criterion F: 0 |
| − | the number of | + | the number of obstructions detected by Tail Reduction: 0 |
| − | the number of redundant generators: | + | the number of obstructions detected by Criterion Bk: 41 |
| + | the number of redundant generators: 5 | ||
It is a partial Groebner basis. | It is a partial Groebner basis. | ||
</example> | </example> | ||
Revision as of 14:54, 9 May 2013
NC.Intersection
Intersection of two finitely generated two-sided ideals in a non-commutative polynomial ring.
Description
Propostion (Intersection of Two Ideals): Let G_I and G_J be two sets of non-zero polynomials in the non-commutative polynomial ring K<x[1],...,x[n]>, and let I and J be two ideals generated by G_I and G_J, respectively. We choose a new indeterminate y, and form the free monoid ring K<y,x[1],...,x[n]>. Furthermore, let N be the ideal generated by the union of {yf: f in G_I} and {(1-y)g: g in G_J}, and let C be the ideal generated by the set {yx[1]-x[1]y,...,yx[n]-x[n]y} of commutators. Then we have the intersection of I and J is equal to the intersection of N+C and K<x[1],...,x[n]>.
Example
-- Let I be the ideal generated by G_I={xy+z,yz+x}, and J be the ideal generated by G_J={yz+x, zx+y}.
-- We compute the intersection of I and J as follows.
Use QQ[t,x,y,z];
NC.SetOrdering("ELIM"); -- Choose an elimination word ordering for t
F1 := [[x,y], [z]]; -- xy+z
F2 := [[y,z], [x]]; -- yz+x
G1 := [[y,z], [x]]; -- yz+x
G2 := [[z,x], [y]]; -- zx+y
N:=[NC.Mul([[t]],F1), NC.Mul([[t]],F2)]; -- t*F1, t*F2
N:=Concat(N,[NC.Mul([[1],[-t]],G1), NC.Mul([[1],[-t]],G2)]); -- (1-t)*G1, (1-t)*G2
C:=[[[t,x],[-x,t]], [[t,y],[-y,t]], [[t,z],[-z,t]]]; -- set of commutators
G:=Concat(N,C);
Gb:=NC.GB(G,31,1,20,50);
-- Done.
-------------------------------
The following information printed by the ApCoCoAServer shows that Gb it is a partial Groebner basis.
the number of unselected generators: 0
the number of unselected obstructions: 81
the procedure is interrupted by loop bound!
the total number of obstructions: 293
the number of selected obstructions: 43
the number of obstructions detected by Criterion M: 128
the number of obstructions detected by Criterion F: 0
the number of obstructions detected by Tail Reduction: 0
the number of obstructions detected by Criterion Bk: 41
the number of redundant generators: 5
It is a partial Groebner basis.
See also
