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NC.MRKernelOfHomomorphism

Enumerate a (partial) Groebner basis of the kernel of a K-algebra homomorphism.
Syntax
          
NC.MRKernelOfHomomorphism(X1:STRING, Relations1:LIST, X2:STRING, Relations2:LIST, Images:LIST):LIST
NC.MRKernelOfHomomorphism(X1:STRING, Relations1:LIST, X2:STRING, Relations2:LIST, Images:LIST, DegreeBound:INT, LoopBound:INT, Flag: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.

Please set coefficient field via the function NC.SetFp (or NC.UnsetFp) before calling this function. The default coefficient field is Q. For more information, please check the relevant function. Since this function is based on Groebner basis computations, we refer users to NC.MRGB or NC.MRReducedGB for information about the following three optional parameters:

Example
-- Example from "On some basic applications of Groebner bases in noncommutative polynomial rings", Patrik Nordbeck
X1 := "abc"; 
Relations1 := []; -- free associative algebra
X2 := "xy"; 
Relations2 := []; -- free associative algebra
F1 := [[1,"x"], [1,"y"]]; 
F2 := [[1,"xx"],[1,"xy"]]; 
F3 := [[1,"yy"],[1,"yx"]]; 
Images := [F1, F2, F3]; -- k-algebra homomorphism is defined by a |->x+y, b |->xx+xy, c |->yy+yx
NC.MRKernelOfHomomorphism(X1, Relations1, X2, Relations2, Images);
[[[1, "ab"], [-1, "ba"], [1, "ac"], [-1, "ca"]], [[1, "aa"], [-1, "b"], [-1, "c"]]]
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-- Small Changes   
F3 := [[1,"yy"],[1,"xy"]]; -- change here 
Images := [F1, F2, F3]; 
NC.MRKernelOfHomomorphism(X1, Relations1, X2, Relations2, Images, 10, 20, 1);
[[[1, "aab"], [-1, "aba"], [1, "aca"], [-1, "caa"]], [[1, "aaa"], [-1, "ab"], [-1, "ca"]], [[1, "abaa"], [-1, "aaca"], [-1, "abb"], [1, "cca"]], 
[[1, "ababa"], [-1, "abbaa"], [-1, "aacab"], [1, "abcaa"], [-1, "aacca"], [1, "ccab"], [1, "ccca"]], 
[[1, "aacaa"], [-1, "abab"], [1, "abba"], [-1, "abca"], [-1, "ccaa"]]] -- it is a partical Groebner basis
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See Also