Difference between revisions of "ApCoCoA-1:NCo.Intersection"

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<command>
 
<command>
 
<title>NCo.Intersection</title>
 
<title>NCo.Intersection</title>
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Intersection of two finitely generated two-sided ideals in a free monoid ring.
 
Intersection of two finitely generated two-sided ideals in a free monoid ring.
 
</short_description>
 
</short_description>
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<syntax></syntax>
 
<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 free nomoid ring <tt>K&lt;x_1,...,x_n&gt;</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&lt;y,x_1,...,x_n&gt;</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_1y,...,yx_n-x_ny}</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&lt;x_1,...,x_n&gt;</tt>.
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<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 free nomoid ring <tt>K&lt;x_1,...,x_n&gt;</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&lt;y,x_1,...,x_n&gt;</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_1y,...,yx_n-x_ny}</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&lt;x_1,...,x_n&gt;</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}.
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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 ObstructionMs: 70
+
the number of unselected MObstructions: 87
 
the procedure is interrupted by loop bound!
 
the procedure is interrupted by loop bound!
the total number of ObstructionMs: 298
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the total number of MObstructions: 349
the number of selected ObstructionMs: 43
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the number of selected MObstructions: 43
the number of ObstructionMs detected by Rule 1: 145
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the number of MObstructions detected by the Criterion M: 162
the number of ObstructionMs detected by Rule 2: 0
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the number of MObstructions detected by the Criterion F: 0
the number of ObstructionMs detected by Rule 3: 40
+
the number of MObstructions detected by the Tail Reduction: 0
 +
the number of MObstructions detected by the Criterion Bk: 57
 
the number of redundant generators: 6
 
the number of redundant generators: 6
 +
A partial Groebner basis of 24 elements is computed.
 
It is a partial Groebner basis.
 
It is a partial Groebner basis.
 
</example>
 
</example>
 
</description>
 
</description>
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<seealso>
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<see>ApCoCoA-1:NCo.FindPolynomials|NCo.FindPolynomials</see>
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<see>ApCoCoA-1:NCo.GB|NCo.GB</see>
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<see>ApCoCoA-1:NCo.Multiply|NCo.Multiply</see>
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<see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see>
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<see>ApCoCoA-1:NCo.SetX|NCo.SetX</see>
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</seealso>
 
<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
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<key>NCo.Intersection</key>
 
<key>NCo.Intersection</key>
 
<key>Intersection</key>
 
<key>Intersection</key>
<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_gbmr</wiki-category>
 
</command>
 
</command>

Latest revision as of 10:20, 7 October 2020

This article is about a function from ApCoCoA-1.

NCo.Intersection

Intersection of two finitely generated two-sided ideals in a free monoid ring.

Syntax

Description

Proposition (Intersection of Two Ideals): Let G_I and G_J be two sets of non-zero polynomials in the free nomoid 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_1y,...,yx_n-x_ny} 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.
NCo.SetX("txyz"); -- Let t be an new indeterminate
NCo.SetOrdering("ELIM"); -- Choose an elimination word ordering for t
F1 := [[1,"xy"], [1,"z"]]; -- xy+z
F2 := [[1,"yz"], [1,"x"]]; -- yz+x
G1 := [[1,"yz"], [1,"x"]]; -- yz+x
G2 := [[1,"zx"], [1,"y"]]; -- zx+y
N:=[NCo.Multiply([[1,"t"]],F1), NCo.Multiply([[1,"t"]],F2)]; -- t*F1, t*F2
N:=Concat(N,[NCo.Multiply([[1,""],[-1,"t"]],G1), NCo.Multiply([[1,""],[-1,"t"]],G2)]); -- (1-t)*G1, (1-t)*G2
C:=[[[1,"tx"],[-1,"xt"]],[[1,"ty"],[-1,"yt"]],[[1,"tz"],[-1,"zt"]]]; -- set of commutators
G:=Concat(N,C); 
Gb:=NCo.GB(G,20,50,1);

-- 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 MObstructions:	87
the procedure is interrupted by loop bound!
the total number of MObstructions:	349
the number of selected MObstructions:	43
the number of MObstructions detected by the Criterion M:	162
the number of MObstructions detected by the Criterion F:	0
the number of MObstructions detected by the Tail Reduction:	0
the number of MObstructions detected by the Criterion Bk:	57
the number of redundant generators:	6
A partial Groebner basis of 24 elements is computed.
It is a partial Groebner basis.

See also

NCo.FindPolynomials

NCo.GB

NCo.Multiply

NCo.SetOrdering

NCo.SetX