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NC.MRReducedGB

Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a finitely presented monoid ring.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term set LT{I} (w.r.t. Ordering) as a monoid ideal. A Groebner basis w.r.t. Ordering is called a reduced Groebner basis w.r.t. Ordering if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. Ordering. However, it is not necessarily finite.
Syntax
          
NC.MRReducedGB(X:STRING, Ordering:STRING, Relations:LIST, G:LIST):LIST
NC.MRReducedGB(X:STRING, Ordering:STRING, Relations:LIST, G: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. About the optional parameters: for most cases we do not know whether or not there exists a finite Groebner basis. Thue, the function has 3 optional parameters to interrupt the enumerating procedure. Note that at the moment all of the following three optional parameters must be used at the same time.

Example
X := "xyzt"; 
Ordering := "LLEX"; 
Relations:=[]; 
F1 := [[1,"xx"], [-1,"yx"]];   
F2 := [[1,"xy"], [-1,"ty"]];  
F3 := [[1,"xt"], [-1, "tx"]];  
F4 := [[1,"yt"], [-1, "ty"]];  
G := [F1, F2,F3,F4]; 
NC.MRGB(X, Ordering, Relations, G);

[[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]], 
[[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]]
-------------------------------
X := "abc"; 
Ordering := "LLEX"; 
Relations := [["aa",""], ["bb",""], ["ab","c"], ["ac", "b"], ["cb", "a"]]; 
F1 := [[1,"a"], [1,"b"], [1,"c"]]; 
F2 := [[1,"b"]]; 
G := [F1, F2];
NC.MRGB(X, Ordering, Relations, G);

[[[1, "b"]], [[1, "a"], [1, "c"]]]
-------------------------------


See Also