# Difference between revisions of "ApCoCoA-1:NCo.BMB"

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

<par/> | <par/> | ||

− | Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>F_{2}<X></tt>, we can consider the <tt>F_{2}</tt>-algebra <tt>F_{2}<X>/I</tt> as a <tt>F_{2}</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt> | + | Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>F_{2}<X></tt>, we can consider the <tt>F_{2}</tt>-algebra <tt>F_{2}<X>/I</tt> as a <tt>F_{2}</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>BLW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>F_{2}</tt>-basis, called a <em>Macaulay's basis</em>, of <tt>F_{2}<X>/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.BHF</ref>) of <tt>F_{2}<X>/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>). |

</short_description> | </short_description> | ||

<syntax> | <syntax> |

## Revision as of 18:26, 30 April 2013

## 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 `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).

### 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>]] -------------------------------

### See also