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

From ApCoCoAWiki
m (fixed links to namespace ApCoCoA)
m (replaced <quotes> tag by real quotes)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
 +
{{Version|1}}
 
<command>
 
<command>
 
<title>NCo.MRInterreduction</title>
 
<title>NCo.MRInterreduction</title>
Line 16: Line 17:
 
<item>@param <em>X:</em> a finite alphabet (or set of indeterminates). It is of STRING type. Note that every letter in X MUST appear only once.The order of letters in X induces word orderings.</item>
 
<item>@param <em>X:</em> a finite alphabet (or set of indeterminates). It is of STRING type. Note that every letter in X MUST appear only once.The order of letters in X induces word orderings.</item>
  
<item>@param <em>Ordering:</em> a word ordering induced by the order of letters in X. It is a STRING, which is a shortened-form of the name of corresponding word ordering. Note that <quotes>LLEX</quotes> (the length-lexicographic ordering), <quotes>ELIM</quotes> (an elimination ordering) and <quotes>LRLEX</quotes> (the length-reverse-lexicographic ordering) are supported currently. See <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref> for more details.</item>
+
<item>@param <em>Ordering:</em> a word ordering induced by the order of letters in X. It is a STRING, which is a shortened-form of the name of corresponding word ordering. Note that "LLEX" (the length-lexicographic ordering), "ELIM" (an elimination ordering) and "LRLEX" (the length-reverse-lexicographic ordering) are supported currently. See <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref> for more details.</item>
  
<item>@param <em>Relations:</em> a finite set of relations. It is of LIST type. Each element in Relations is of the form [W1, W2], where W1 and W2 are words in <tt>&lt;X&gt;</tt>. Each word is represented as a STRING. For example, the word <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, and the identity is represented as the empty string <quotes></quotes>. Thus, the relation <tt>(yx, xy)</tt> is represented as [<quotes>yx</quotes>, <quotes>xy</quotes>], and the set of relations <tt>{(yx, xy),(zx,xz),(zy,yz)}</tt> is represented as [[<quotes>yx</quotes>, <quotes>xy</quotes>],[<quotes>zx</quotes>, <quotes>xz</quotes>],[<quotes>zy</quotes>, <quotes>yz</quotes>]].</item>
+
<item>@param <em>Relations:</em> a finite set of relations. It is of LIST type. Each element in Relations is of the form [W1, W2], where W1 and W2 are words in <tt>&lt;X&gt;</tt>. Each word is represented as a STRING. For example, the word <tt>xy^2x</tt> is represented as "xyyx", and the identity is represented as the empty string "". Thus, the relation <tt>(yx, xy)</tt> is represented as ["yx", "xy"], and the set of relations <tt>{(yx, xy),(zx,xz),(zy,yz)}</tt> is represented as [["yx", "xy"],["zx", "xz"],["zy", "yz"]].</item>
  
<item>@param <em>G:</em> a LIST of polynomials in the finitely generated monoid ring. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
+
<item>@param <em>G:</em> a LIST of polynomials in the finitely generated monoid ring. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
  
 
<item>@return: a LIST of interreduced polynomials.</item>
 
<item>@return: a LIST of interreduced polynomials.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
X := <quotes>abc</quotes>;
+
X := "abc";
Ordering := <quotes>LLEX</quotes>;
+
Ordering := "LLEX";
Relations := [[<quotes>aa</quotes>,<quotes></quotes>], [<quotes>bb</quotes>,<quotes></quotes>], [<quotes>ab</quotes>,<quotes>c</quotes>], [<quotes>ac</quotes>, <quotes>b</quotes>], [<quotes>cb</quotes>, <quotes>a</quotes>]];
+
Relations := [["aa",""], ["bb",""], ["ab","c"], ["ac", "b"], ["cb", "a"]];
G:=[[[1,<quotes>ba</quotes>]], [[1,<quotes>b</quotes>],[1,<quotes></quotes>]], [[1,<quotes>c</quotes>]]];
+
G:=[[[1,"ba"]], [[1,"b"],[1,""]], [[1,"c"]]];
 
NCo.MRInterreduction(X, Ordering, Relations, G);
 
NCo.MRInterreduction(X, Ordering, Relations, G);
  
[[[1, <quotes>a</quotes>]], [[1, <quotes>b</quotes>], [1, <quotes></quotes>]], [[1, <quotes>c</quotes>]]]
+
[[[1, "a"]], [[1, "b"], [1, ""]], [[1, "c"]]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>

Latest revision as of 13:41, 29 October 2020

This article is about a function from ApCoCoA-1.

NCo.MRInterreduction

Interreduce a LIST of polynomials in a finitely presented monoid ring.

Syntax

NCo.MRInterreduction(X:STRING, Ordering:STRING, Relations:LIST, G:LIST):LIST

Description

Given a word ordering, a set G of non-zero polynomials is called interreduced if no element of Supp(g) is a multiply of any element in the leading word set MRLW{G\{g}} for all g in G.

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 coefficient field via the function NCo.SetFp (or NCo.UnsetFp) before calling this function. The default coefficient field is the field of rational numbers, i.e. RAT in CoCoAL. For more information, please check the relevant functions.

  • @param X: a finite alphabet (or set of indeterminates). It is of STRING type. Note that every letter in X MUST appear only once.The order of letters in X induces word orderings.

  • @param Ordering: a word ordering induced by the order of letters in X. It is a STRING, which is a shortened-form of the name of corresponding word ordering. Note that "LLEX" (the length-lexicographic ordering), "ELIM" (an elimination ordering) and "LRLEX" (the length-reverse-lexicographic ordering) are supported currently. See NCo.SetOrdering for more details.

  • @param Relations: a finite set of relations. It is of LIST type. Each element in Relations is of the form [W1, W2], where W1 and W2 are words in <X>. Each word is represented as a STRING. For example, the word xy^2x is represented as "xyyx", and the identity is represented as the empty string "". Thus, the relation (yx, xy) is represented as ["yx", "xy"], and the set of relations {(yx, xy),(zx,xz),(zy,yz)} is represented as [["yx", "xy"],["zx", "xz"],["zy", "yz"]].

  • @param G: a LIST of polynomials in the finitely generated monoid ring. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial 0 is represented as the empty LIST [].

  • @return: a LIST of interreduced polynomials.

Example

X := "abc";
Ordering := "LLEX";
Relations := [["aa",""], ["bb",""], ["ab","c"], ["ac", "b"], ["cb", "a"]];
G:=[[[1,"ba"]], [[1,"b"],[1,""]], [[1,"c"]]];
NCo.MRInterreduction(X, Ordering, Relations, G);

[[[1, "a"]], [[1, "b"], [1, ""]], [[1, "c"]]]
-------------------------------

See also

NCo.MRLW

NCo.SetFp

NCo.SetOrdering

NCo.UnsetFp

Introduction to CoCoAServer