Difference between revisions of "ApCoCoA-1:NCo.BMB"
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| − | 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>). | + | 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>ApCoCoA-1:NCo.BHF|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>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>). |
<par/> | <par/> | ||
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||
<par/> | <par/> | ||
| − | Please set ring environment <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before calling this function. The default ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions. | + | Please set ring environment <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before calling this function. The default ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions. |
<itemize> | <itemize> | ||
<item>@param <em>Gb:</em> a LIST of non-zero polynomials in the free monoid ring <tt>F_{2}<X></tt> 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 <tt><X></tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, and the identity is represented as the empty string <quotes></quotes>. Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[<quotes>xy</quotes>, <quotes>y</quotes>, <quotes></quotes>]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> 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.</item> | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in the free monoid ring <tt>F_{2}<X></tt> 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 <tt><X></tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, and the identity is represented as the empty string <quotes></quotes>. Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[<quotes>xy</quotes>, <quotes>y</quotes>, <quotes></quotes>]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> 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.</item> | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>NCo.BHF</see> | + | <see>ApCoCoA-1:NCo.BHF|NCo.BHF</see> |
| − | <see>NCo.BIsGB</see> | + | <see>ApCoCoA-1:NCo.BIsGB|NCo.BIsGB</see> |
| − | <see>NCo.BLW</see> | + | <see>ApCoCoA-1:NCo.BLW|NCo.BLW</see> |
| − | <see>NCo.SetOrdering</see> | + | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> |
| − | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
Revision as of 08:27, 7 October 2020
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(<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
