# ApCoCoA-1:NCo.BMB

## NCo.BMB

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

### Syntax

```NCo.BMB(Gb:LIST[, DB:INT]):LIST
```

### Description

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 BLW{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).

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("xyzt");
NCo.SetOrdering("LLEX");
GB:= [["yt", "ty"], ["xt", "tx"], ["xy", "ty"], [ "xx", "yx"],
["tyy", "tty"], ["yyx", "tyx"]];
NCo.BMB(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"]]
-------------------------------
```