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NCo.MB

Enumerate a Macaulay's basis of a finitely generated K-algebra.
Syntax
          
NCo.MB(Gb:LIST[, DB:INT]):LIST

          

Description
Given a two-sided ideal I in a free monoid ring K, we can consider the K-algebra K/I as a K-vector space. Moreover, let G be a Groebner basis of I, and let B be the set of all words which are not a multiple of any word in the leading word set LW{G}. Then the residue class of the words in B form a K-basis, called a Macaulay's basis, of K/I. For the sake of computing the values of the Hilbert function (see NCo.HF) of K/I, in this function we require that G has to be a Groebner basis with respect to a length compatible word ordering (see NCo.SetOrdering).

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 ring environment coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is Q, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions. Optional parameter:

Example
NCo.SetX("xyzt");
NCo.SetOrdering("LLEX");
Gb:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]], 
[[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]];
NCo.MB(Gb,3);
[[""], ["t", "z", "y", "x"], ["tt", "tz", "ty", "tx", "zt", "zz", "zy", "zx", "yz", "yy", "yx", "xz"], 
["ttt", "ttz", "tty", "ttx", "tzt", "tzz", "tzy", "tzx", "tyz", "tyx", "txz", "ztt", "ztz", "zty", "ztx",
"zzt", "zzz", "zzy", "zzx", "zyz", "zyy", "zyx", "zxz", "yzt", "yzz", "yzy", "yzx", "yyz", "yyy",
 "yxz", "xzt", "xzz", "xzy", "xzx"]]
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See Also