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

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{{Version|1}}
 
<command>
 
<command>
 
<title>NCo.BMB</title>
 
<title>NCo.BMB</title>
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</syntax>
 
</syntax>
 
<description>
 
<description>
Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>F_{2}&lt;X&gt;</tt>, we can consider the <tt>F_{2}</tt>-algebra <tt>F_{2}&lt;X&gt;/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}&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.BHF</ref>) of <tt>F_{2}&lt;X&gt;/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>).
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Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>F_{2}&lt;X&gt;</tt>, we can consider the <tt>F_{2}</tt>-algebra <tt>F_{2}&lt;X&gt;/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}&lt;X&gt;/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}&lt;X&gt;/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.
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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 ("LLEX"). 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}&lt;X&gt;</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>&lt;X&gt;</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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<item>@param <em>Gb:</em> a LIST of non-zero polynomials in the free monoid ring <tt>F_{2}&lt;X&gt;</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>&lt;X&gt;</tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=["xy", "y", ""]. 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>@return: a LIST of words forming a Macaulay's basis of the <tt>F_{2}</tt>-algebra <tt>F_{2}&lt;X&gt;/&lt;Gb&gt;</tt>.</item>
 
<item>@return: a LIST of words forming a Macaulay's basis of the <tt>F_{2}</tt>-algebra <tt>F_{2}&lt;X&gt;/&lt;Gb&gt;</tt>.</item>
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</itemize>
 
</itemize>
 
<example>
 
<example>
NCo.SetX(<quotes>xyzt</quotes>);
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NCo.SetX("xyzt");
NCo.SetOrdering(<quotes>LLEX</quotes>);
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NCo.SetOrdering("LLEX");
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>],  
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GB:= [["yt", "ty"], ["xt", "tx"], ["xy", "ty"], [ "xx", "yx"],  
[<quotes>tyy</quotes>, <quotes>tty</quotes>], [<quotes>yyx</quotes>, <quotes>tyx</quotes>]];
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["tyy", "tty"], ["yyx", "tyx"]];
 
NCo.BMB(GB,3);
 
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>],  
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[[""], ["t", "z", "y", "x"], ["tt", "tz", "ty", "tx", "zt", "zz", "zy", "zx", "yz", "yy", "yx", "xz"],  
[<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>,  
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["ttt", "ttz", "tty", "ttx", "tzt", "tzz", "tzy", "tzx", "tyz", "tyx", "txz", "ztt", "ztz", "zty", "ztx",  
<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>,  
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"zzt", "zzz", "zzy", "zzx", "zyz", "zyy", "zyx", "zxz", "yzt", "yzz", "yzy", "yzx", "yyz", "yyy", "yxz",  
<quotes>xzt</quotes>, <quotes>xzz</quotes>, <quotes>xzy</quotes>, <quotes>xzx</quotes>]]
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"xzt", "xzz", "xzy", "xzx"]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NCo.BHF</see>
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<see>ApCoCoA-1:NCo.BHF|NCo.BHF</see>
<see>NCo.BIsGB</see>
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<see>ApCoCoA-1:NCo.BIsGB|NCo.BIsGB</see>
<see>NCo.BLW</see>
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<see>ApCoCoA-1:NCo.BLW|NCo.BLW</see>
<see>NCo.SetOrdering</see>
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<see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see>
<see>NCo.SetX</see>
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<see>ApCoCoA-1:NCo.SetX|NCo.SetX</see>
<see>Introduction to CoCoAServer</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>
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<key>NCo.BMB</key>
 
<key>NCo.BMB</key>
 
<key>BMB</key>
 
<key>BMB</key>
<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_gbmr</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:38, 29 October 2020

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

NCo.BHF

NCo.BIsGB

NCo.BLW

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer