# ApCoCoA-1:NCo.BMB

This article is about a function from ApCoCoA-1. |

## 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"]] -------------------------------

### See also