Difference between revisions of "ApCoCoA-1:NC.KernelOfHomomorphism"
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(New page: <command> <title>NC.KernelOfHomomorphism</title> <short_description> Computing (patial) Groebner basis of kernel of k-algebra homomorphism. </short_description> <syntax> NC.KernelOfHomomor...) |
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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. | <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. | ||
<itemize> | <itemize> | ||
| − | <item> | + | <item>Before calling the function, please set coefficient field <tt>K</tt> through the functions NC.SetFp(Prime) (or NC.UnsetFp()). Default coefficient field is <tt>Q</tt>. For more information, please check the relevant functions.</item> |
| − | <item>@param <em> | + | <item>@param <em>X_1:</em> a STRING which is an alphabet (indeterminates) of a free associative K-algebra. Every letter in <tt>X_1</tt> should occurrence only once. Note that the order of letters in <tt>X_1</tt> is important since it induces an admissible ordering and defines a <tt>K</tt>-homomorphism as well.</item> |
| − | <item>@param <em>X_2:</em> another | + | <item>@param <em>X_2:</em> a STRING which is another alphabet (indeterminates) of another free associative <tt>K</tt>-algebra.</item> |
| − | <item>@param <em> | + | <item>@param <em>Images:</em>a LIST of polynomials generating a two-sided ideal in <tt>K<X_2></tt>. Each polynomial in <tt>K<X_2></tt> is represented as a LIST of LISTs, which are pairs of form [c, w] where c is in <tt>K</tt> and w is a word in <tt>(X_2)*</tt>. Unit in <tt>(X_2)*</tt> is empty word represented as an empty STRING "". <tt>0</tt> polynomial is represented as an empty LIST []. For example, X_2:=<quotes>xy</quotes>; F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]; means polynomial <tt>F:=xy-y+1</tt> in <tt>K<x,y></tt>. Note that the order of polynomials in <tt>Images</tt> is important since it defines a <tt>K</tt>-homomorphism.</item> |
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</itemize> | </itemize> | ||
| + | Since the algorithm used in this function is based on Groebner basis computation, we refer users to <tt>NC.GB</tt> for information on optional parameters. | ||
<example> | <example> | ||
X_1 := <quotes>abc</quotes>; | X_1 := <quotes>abc</quotes>; | ||
| Line 36: | Line 29: | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| + | <see>NC.Add</see> | ||
| + | <see>NC.GB</see> | ||
| + | <see>NC.Intersection</see> | ||
| + | <see>NC.IsGB</see> | ||
| + | <see>NC.LC</see> | ||
| + | <see>NC.LT</see> | ||
| + | <see>NC.LTIdeal</see> | ||
| + | <see>NC.MinimalPolynomial</see> | ||
| + | <see>NC.Multiply</see> | ||
| + | <see>NC.NR</see> | ||
| + | <see>NC.SetFp</see> | ||
| + | <see>NC.SetOrdering</see> | ||
| + | <see>NC.SetRelations</see> | ||
| + | <see>NC.SetRules</see> | ||
| + | <see>NC.SetX</see> | ||
| + | <see>NC.Subtract</see> | ||
| + | <see>NC.UnsetFp</see> | ||
| + | <see>NC.UnsetOrdering</see> | ||
| + | <see>NC.UnsetRelations</see> | ||
| + | <see>NC.UnsetRules</see> | ||
| + | <see>NC.UnsetX</see> | ||
<see>Gbmr.MRAdd</see> | <see>Gbmr.MRAdd</see> | ||
<see>Gbmr.MRBP</see> | <see>Gbmr.MRBP</see> | ||
<see>Gbmr.MRIntersection</see> | <see>Gbmr.MRIntersection</see> | ||
| + | <see>Gbmr.MRKernelOfHomomorphism</see> | ||
<see>Gbmr.MRMinimalPolynomials</see> | <see>Gbmr.MRMinimalPolynomials</see> | ||
<see>Gbmr.MRMultiply</see> | <see>Gbmr.MRMultiply</see> | ||
Revision as of 09:48, 20 July 2010
NC.KernelOfHomomorphism
Computing (patial) Groebner basis of kernel of k-algebra homomorphism.
Syntax
NC.KernelOfHomomorphism(X_1:STRING, X_2:STRING, Images:LIST[, DegreeBound:INT, LoopBound:INT, IFlag:INT]):LIST
Description
Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.
Before calling the function, please set coefficient field K through the functions NC.SetFp(Prime) (or NC.UnsetFp()). Default coefficient field is Q. For more information, please check the relevant functions.
@param X_1: a STRING which is an alphabet (indeterminates) of a free associative K-algebra. Every letter in X_1 should occurrence only once. Note that the order of letters in X_1 is important since it induces an admissible ordering and defines a K-homomorphism as well.
@param X_2: a STRING which is another alphabet (indeterminates) of another free associative K-algebra.
@param Images:a LIST of polynomials generating a two-sided ideal in K<X_2>. Each polynomial in K<X_2> is represented as a LIST of LISTs, which are pairs of form [c, w] where c is in K and w is a word in (X_2)*. Unit in (X_2)* is empty word represented as an empty STRING "". 0 polynomial is represented as an empty LIST []. For example, X_2:="xy"; F:=[[1,"xy"], [-1, "y"], [1,""]]; means polynomial F:=xy-y+1 in K<x,y>. Note that the order of polynomials in Images is important since it defines a K-homomorphism.
Since the algorithm used in this function is based on Groebner basis computation, we refer users to NC.GB for information on optional parameters.
Example
X_1 := <quotes>abc</quotes>; X_2 := <quotes>xy</quotes>; F1 := [[1,<quotes>x</quotes>], [1,<quotes>y</quotes>]]; F2 := [[1,<quotes>xx</quotes>],[1,<quotes>xy</quotes>]]; F3 := [[1,<quotes>yy</quotes>],[1,<quotes>yx</quotes>]]; Images :=[F1, F2, F3]; -- a |-> F1; b |-> F2; c |-> F3 NC.KernelOfHomomorphism(X_1, X_2, Images); [[[1, <quotes>ab</quotes>], [-1, <quotes>ba</quotes>], [1, <quotes>ac</quotes>], [-1, <quotes>ca</quotes>]], [[1, <quotes>aa</quotes>], [-1, <quotes>b</quotes>], [-1, <quotes>c</quotes>]]] -------------------------------
See also
