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

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.Intersection</title>
 
<title>NC.Intersection</title>
 
<short_description>
 
<short_description>
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Enumerate a (partial) Groebner basis of the intersection of two finitely generated two-sided ideals over a free monoid ring.
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Intersection of two finitely generated two-sided ideals in a non-commutative polynomial ring.
 
</short_description>
 
</short_description>
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<syntax>
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<syntax></syntax>
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NC.Intersection(G1:LIST, G2:LIST):LIST
 
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NC.Intersection(G1:LIST, G2:LIST, DegreeBound:INT, LoopBound:INT, Flag:INT):LIST
 
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</syntax>
 
 
<description>
 
<description>
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<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
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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 non-commutative polynomial 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[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&lt;x[1],...,x[n]&gt;</tt>.
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<par/>
 
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Please set ring environment <em>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.SetOrdering</ref>, respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
 
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<itemize>
 
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<item>@param <em>G1, G2:</em> two LISTs of non-zero polynomials which generate two two-sided ideals in <tt>K&lt;X&gt;</tt>. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]].</item>
 
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</itemize>
 
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Since this function is based on Groebner basis computations, we refer users to <ref>NC.GB</ref> or <ref>NC.ReducedGB</ref> for information about the following three optional parameters:
 
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<itemize>
 
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<item>@param <em>DegreeBound</em></item>
 
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<item>@param <em>LoopBound</em></item>
 
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<item>@param <em>Flag</em></item>
 
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<item>@return: a LIST of polynomials, which is a Groebner basis of the intersection of the two-sided ideals <tt>(G1)</tt> and <tt>(G2)</tt> if a finite Groebner basis exists, and is a partial Groebner basis otherwise.</item>
 
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</itemize>
 
 
<example>
 
<example>
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NC.SetFp(); -- set default Fp=F2
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-- 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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NC.SetX(<quotes>xyz</quotes>);  
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-- We compute the intersection of I and J as follows.
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F1 := [[1,<quotes>xy</quotes>], [1,<quotes>z</quotes>]];  
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Use QQ[t,x,y,z];
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F2 := [[1,<quotes>yz</quotes>], [1, <quotes>x</quotes>]];  
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NC.SetOrdering("ELIM"); -- Choose an elimination word ordering for t
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F3 := [[1,<quotes>zx</quotes>], [1,<quotes>y</quotes>]];
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F1 := [[x,y], [z]]; -- xy+z
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G1 := [F1, F2]; -- ideal generated by {xy+z, yz+x}
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F2 := [[y,z], [x]]; -- yz+x
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G2 := [F2, F3]; -- ideal generated by {yz+x, zx+y}
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G1 := [[y,z], [x]]; -- yz+x
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NC.Intersection(G1, G2, 20, 25, 1);
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G2 := [[z,x], [y]]; -- zx+y
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[[[1, <quotes>yzyz</quotes>], [1, <quotes>zyzy</quotes>]], [[1, <quotes>zzyzyy</quotes>], [1, <quotes>yyzy</quotes>], [1, <quotes>zyzz</quotes>], [1, <quotes>yz</quotes>]],  
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N:=[NC.Mul([[t]],F1), NC.Mul([[t]],F2)]; -- t*F1, t*F2
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[[1, <quotes>yzzyzy</quotes>], [1, <quotes>yzyy</quotes>], [1, <quotes>zzyz</quotes>], [1, <quotes>zy</quotes>]], [[1, <quotes>x</quotes>], [1, <quotes>yz</quotes>]]]
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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
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G:=Concat(N,C);
 +
Gb:=NC.GB(G,31,1,20,50);
 +
 
 +
-- Done.
 
-------------------------------
 
-------------------------------
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Note the following information printed by the server shows it is a partial Groebner basis.  
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===== 25th Loop =====
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Number of elements in (partial) Groebner basis G: 22 -- partial Groebner basis before being interreduced
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The following information printed by the ApCoCoAServer shows that Gb it is a partial Groebner basis.  
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Number of S-elements: 25/86 -- 25 S-elements have been check, and 61 (=86-25) unchecked S-elements
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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.
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
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<see>NC.Add</see>
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<see>ApCoCoA-1:Use|Use</see>
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<see>NC.Deg</see>
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<see>ApCoCoA-1:NC.FindPolys|NC.FindPolys</see>
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<see>NC.FindPolynomials</see>
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<see>ApCoCoA-1:NC.GB|NC.GB</see>
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<see>NC.GB</see>
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<see>ApCoCoA-1:NC.Mul|NC.Mul</see>
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<see>NC.HF</see>
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<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
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<see>NC.Interreduction</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
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<see>NC.Intersection</see>
 
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<see>NC.IsGB</see>
 
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<see>NC.KernelOfHomomorphism</see>
 
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<see>NC.LC</see>
 
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<see>NC.LT</see>
 
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<see>NC.LTIdeal</see>
 
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<see>NC.MinimalPolynomial</see>
 
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<see>NC.Multiply</see>
 
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<see>NC.NR</see>
 
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<see>NC.ReducedGB</see>
 
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<see>NC.SetFp</see>
 
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<see>NC.SetOrdering</see>
 
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<see>NC.SetRelations</see>
 
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<see>NC.SetRules</see>
 
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<see>NC.SetX</see>
 
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<see>NC.Subtract</see>
 
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<see>NC.UnsetFp</see>
 
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<see>NC.UnsetOrdering</see>
 
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<see>NC.UnsetRelations</see>
 
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<see>NC.UnsetRules</see>
 
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<see>NC.UnsetX</see>
 
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<see>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>ideal</type>
 
 
<type>non_commutative</type>
 
<type>non_commutative</type>
 
</types>
 
</types>
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<key>gbmr.Intersection</key>
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<key>ncpoly.Intersection</key>
 
<key>NC.Intersection</key>
 
<key>NC.Intersection</key>
 
<key>Intersection</key>
 
<key>Intersection</key>
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<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

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

Use

NC.FindPolys

NC.GB

NC.Mul

NC.SetOrdering

Introduction to CoCoAServer




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