# ApCoCoA-1:NCo.BMB

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## NCo.BMB

Enumerate a Macauley's basis of a finitely generated algebra over the binary field F_{2}={0,1}.

Given a two-sided ideal I in a free monoid ring F_{2}<X>, we can consider the F_{2}-algebra F_{2}<X>/I as a F_{2}-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 F_{2}-basis, called a Macaulay's basis, of F_{2}<X>/I. For the sake of computing the values of the Hilbert function (see NCo.BHF) of F_{2}<X>/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).

### Syntax

```NCo.BMB(Gb:LIST[, DB: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 ring environment alphabet (or set of indeterminates) X and word ordering via the functions NCo.SetX and NCo.SetOrdering, respectively, before calling this function. The default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

• @param Gb: a LIST of non-zero polynomials in the free monoid ring F_{2}<X> which is a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of words (or terms) in <X>. Each word is represented as a STRING. For example, xy^2x is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial f=xy-y+1 is represented as F:=["xy", "y", ""]. The zero polynomial 0 is represented as the empty LIST []. Warning: users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.

• @return: a LIST of words forming a Macaulay's basis of the F_{2}-algebra F_{2}<X>/<Gb>.

Optional parameter:

• @param DB: a positive INT which is a degree bound of the lengths of words. Note that we set DB=32 by default. Thus, in the case that F_{2}<X>/<Gb> has a finite Macaulay's basis, it is necessary to set DB to a large enough INT in order to compute the whole Macaulay's basis.

#### Example

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