Difference between revisions of "ApCoCoA-1:NC.Intersection"
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Intersection of two finitely generated two-sided ideals in a non-commutative polynomial ring. | Intersection of two finitely generated two-sided ideals in a non-commutative polynomial ring. | ||
</short_description> | </short_description> | ||
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<description> | <description> | ||
<em>Proposition (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>. | <em>Proposition (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>. | ||
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<seealso> | <seealso> | ||
| − | <see>NC.FindPolys</see> | + | <see>ApCoCoA-1:Use|Use</see> |
| − | <see>NC.GB</see> | + | <see>ApCoCoA-1:NC.FindPolys|NC.FindPolys</see> |
| − | <see>NC.Mul</see> | + | <see>ApCoCoA-1:NC.GB|NC.GB</see> |
| − | <see>NC.SetOrdering</see> | + | <see>ApCoCoA-1:NC.Mul|NC.Mul</see> |
| + | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> | ||
| + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> | ||
</seealso> | </seealso> | ||
<types> | <types> | ||
<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||
| + | <type>ideal</type> | ||
<type>groebner</type> | <type>groebner</type> | ||
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<type>non_commutative</type> | <type>non_commutative</type> | ||
</types> | </types> | ||
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<key>Intersection</key> | <key>Intersection</key> | ||
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Latest revision as of 13:34, 29 October 2020
| This article is about a function from ApCoCoA-1. |
NC.Intersection
Intersection of two finitely generated two-sided ideals in a non-commutative polynomial ring.
Syntax
Description
Proposition (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
