Difference between revisions of "ApCoCoA-1:NCo.Intersection"
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<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> | ||
| + | <syntax></syntax> | ||
<description> | <description> | ||
| − | <em> | + | <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<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_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<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}. | ||
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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 MObstructions: 87 |
the procedure is interrupted by loop bound! | the procedure is interrupted by loop bound! | ||
| − | the total number of | + | the total number of MObstructions: 349 |
| − | the number of selected | + | the number of selected MObstructions: 43 |
| − | the number of | + | the number of MObstructions detected by the Criterion M: 162 |
| − | the number of | + | the number of MObstructions detected by the Criterion F: 0 |
| − | the number of | + | 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> | ||
<seealso> | <seealso> | ||
| − | <see>NCo.FindPolynomials</see> | + | <see>ApCoCoA-1:NCo.FindPolynomials|NCo.FindPolynomials</see> |
| − | <see>NCo.GB</see> | + | <see>ApCoCoA-1:NCo.GB|NCo.GB</see> |
| − | <see>NCo.Multiply</see> | + | <see>ApCoCoA-1:NCo.Multiply|NCo.Multiply</see> |
| − | <see>NCo.SetOrdering</see> | + | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> |
| − | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
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<key>NCo.Intersection</key> | <key>NCo.Intersection</key> | ||
<key>Intersection</key> | <key>Intersection</key> | ||
| − | <wiki-category>Package_gbmr</wiki-category> | + | <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
