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Package gbmr is designed to enable us to do basic operations (for instance addition, subtraction, multiplication, normal remainder, leading term, etc.) over Non-Commutative algebra, i.e. finitely presented monoid rings, compute (partial) Groebner bases of finitely generated (one-sided/two-sided) ideals, and experiment on applications of Groebner bases.
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The package gbmr contains numbers of functions for basic computations and Groebner basis computations in <em>non-commutative algebras</em>, such as finitely generated free monoid rings (or non-commutative polynomial rings, non-commutative free associative algebras), finitely presented monoid rings, group ring, etc., over the field of rational numbers Q or over finite fields Z/(p) where p is a prime. More precisedly, this package enables us to do computations as addition, subtraction and multiplication of two non-commutative polynomials, getting the leading word and leading coefficient of a non-zero polynomial, computing the normal remainder of a polynomial w.r.t. a list of polynomials, interreducing a lists of polynomials, enumerating (reduced) (partial) Groebner bases of finitely generated two-sided ideals, and computing truncated Groebner basis of a finitely and homogeneously generated two-sided ideals, etc. Consequently, this package can be applied to many algebraic applications, for instance, enumerating a Macaulay's basis and the values of the Hilbert function of a finitely generated K-algbera, computing leading word ideals, intersections of ideals, and kernels of K-algebra homomorphisms, and so on.
  
Generally, a finitely presented monoid ring is defined by <tt>P=K<X|R>=K<X>/<R></tt>, where <tt>K</tt> is a field, <tt>X</tt> is a finite alphabet or a finite set of letters, and <tt>R</tt> is a finite set of relations. If <tt>R</tt> is empty, then <tt>P</tt> becomes a free associative <tt>K</tt>-algebra.
 
  
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Generally speaking, a finitely presented monoid ring is defined by <tt>P=K<X|R></tt>, where <tt>K</tt> is a field, <tt>X</tt> is a finite alphabet (or a finite set of indeterminates), and <tt>R</tt> is a finite set of relations. Clearly, we have <tt>P=K<X|R></tt> is isomorphic to <tt>K<X>/<R></tt>, where <tt>K<X></tt> is the free monoid ring generated by <tt>X</tt> over <tt>K</tt> and <tt><R></tt> is the two-sided ideal generated by <tt>R</tt>.
  
Things to know about this package.
 
  
(a) Predefined alias for this package is
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<strong>Important issues about this package:</strong>
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Alias NC := $apcocoa/gbmr;
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(a) Predefined alias for this package is as follows.
  
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Alias NCo := $apcocoa/gbmr;
  
(b) <tt>K</tt> is field of rational number <tt>Q</tt> by default. It can be set to a finite field <tt>Fp</tt> through the functions
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Note that, before ApCoCoA 1.9.0,  the alias for this package is NC. However, since ApCoCoA 1.9.0, the alias NC has been used for the ApCoCoA package ncpoly.
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetFp(); and NC.SetFp(Prime);
 
  
where <tt>Prime</tt>is a prime number. The prevouse one sets finite field to <tt>F_{2}=Z/(2)</tt> and the later to <tt>F_{Prime}=Z/(Prime)</tt>. And <tt>K</tt> can be reset to field of rational number through the function
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(b) By default, the field <tt>K</tt> is the field of rational numbers. It can be set to a finite field through the functions
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetFp();
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetFp(); and NCo.SetFp(P);
  
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where <tt>P</tt> is a prime number. The former sets the finite field to the binary field <tt>{0,1}</tt>, and the latter to the finite field <tt>{0,1,2,...P-1}</tt>. One can reset <tt>K</tt> to rational numbers via the function
  
(c) <tt>X</tt> (or Alphabet) is represented as a STRING of letters. Every letter in <tt>X</tt> should have a unique occurrence. The order of letters in <tt>X</tt> is important since it induces an admissible ordering specified by Ordering. <tt>X</tt> can be set through the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.UnsetFp();
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetX(X);
 
  
where <tt>X</tt> is a STRING of letters. And <tt>X</tt> can be reset to empty through the function
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(c) The alphabet <tt>X</tt> is represented as a STRING of letters. Every letter in <tt>X</tt> must have a unique appearance. The order of letters in <tt>X</tt> is important since it will induce word orderings on the free monoid <tt><X></tt> (see NCo.SetOrdering). The alphabet <tt>X</tt> is set via the function
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetX();
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetX(X);
  
However I fail to find a proper situation to use it currently.
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where <tt>X</tt> is a STRING of letters.
  
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (c.1) Each word (term) in the free monoid <tt><X></tt> is represented as a STRING with all letters coming from <tt>X</tt>. For example, the word
  
(d) Ordering is a STRING indicating which ordering we are working with. In the package we use admissible orderings. Currently, the package only supports length-lexicographic ordering ("LLEX") and elimination ordering ("ELIM") induced from the order of letters in <tt>X</tt>. The default ordering is "LLEX".
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>w=ba</tt>  
  
For example, X:="abc"; Ordering:="ELIM"; means elimination ordering induced from <tt>a>b>c</tt>.
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is represented as
  
Ordering can be set through the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; W:="ba";
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetOrdering(Ordering);
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Note that the identity element in <tt><X></tt> is the empty word which is represented as the empty STRING "".
  
where Ordering is the ordering supported by the package. And Ordering can be reset to  "LLEX" through the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (c.2) Each non-commutative polynomial is represented as a LIST of monomials, and each monomial is represented as a LIST consisting of an element (coefficient) in <tt>K</tt> and a word (term) in <tt><X></tt>. For example the polynomial
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetOrdering();
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>f=ab+2b^2+3</tt>
  
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is represented as
  
(e) Relations, which is a finite generating set, is represented as a LIST of relations. Each relation of Relations is represented as a LIST (pair) composting of two words in <tt>X*</tt>.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  F := [[1,"ab"],[2,"bb"],[3,""]];
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(e0) Each word (term) in <tt>X*</tt> is represented as a STRING with all letters coming from <tt>X</tt>.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Note that the zero polynomial <tt>0</tt> is represented as the empty LIST [].  
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;For example, X := "abc"; W := "ba"; means <tt>W=ba</tt>.  
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (c.3) In the case that <tt>K={0,1}</tt>, every polynomial can be represented as a LIST of words (terms) in <tt><X></tt>. For example, the polynomial
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Note that unit in <tt>X*</tt> is empty word represented as an empty STRING "".
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>p=ab+b^2+1</tt>  
  
For example, X := "abc"; Relations := [["ba","ab"], ["ca","ac"], ["cb","bc"]]; means Relations generated by <tt>{ba=ab, ca=ac, cb=bc}</tt>.
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is represented as
  
Relations can be set through the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; P := ["ab","bb",""];
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetRelations(Relations);
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Notice that this representation is ONLY applied to computations in free monoid rings over the binary field <tt>{0,1}</tt>. See functions with the prefix "B" for more details.
  
where Relations is a properly represented Relations. And Relations can be reset to empty through the function
 
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetRelations();
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(d) A <em>word ordering</em> on a monoid is a well-ordering that is compatible with multiplication. One can set word orderings via the function
  
which might be a tricky way to change a monoid ring to a free associative <tt>K</tt>-algebra.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetOrdering(Ordering);
  
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where <tt>Ordering</tt> is a STRING indicating which ordering we are going to work with. Currently, the package only supports the length-lexicographic ordering ("LLEX"), an elimination ordering ("ELIM") and the length-reverse-lexicographic ordering ("LRLEX"). We refer to NCo.SetOrdering for the definitions of these orderings. The default ordering is "LLEX". Note that word orderings are induced by the order of letters in <tt>X</tt>. For example, X:="abc"; Ordering:="LLEX"; means the length-lexicographic word ordering induced by <tt>a>b>c</tt>.
  
(f) Rules, which is also a finite generating set, is represented as a LIST of (rewriting) rules. Each rule of Rules is represented as a LIST (pair) consisting of one word in <tt>X*</tt> and one polynomial in <tt>K<X|R></tt>.
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(e) For a finitely presented monoid ring <tt>P=K<X|R></tt>, the set <tt>R</tt> of relations is represented as a LIST. and each relation in <tt>R</tt> is represented as a LIST composed of two words in <tt><X></tt>. For example, the relations
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(f0) Each polynomial in <tt>K<X|R></tt> is represented as a LIST of monomials, and each monomial is represented as a LIST (pair) consisting of one coefficient in <tt>K</tt> and one word (term) in <tt>X*</tt>.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>R={ba=ab, ca=ac, cb=bc}</tt>
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;For example, X := "abc"; P := [[1,"ab"], [1,""]]; means <tt>P=ab+1</tt>.
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is represented as
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Note that 0 polynomial is represented as an empty LIST [].
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; R:= [["ba","ab"], ["ca","ac"], ["cb","bc"]];
  
For example, X := "ab"; Rules := [["ba",  [[1,"ab"], [1,""]]]]; means Rules generated by <tt>{ba=ab+1}</tt>.
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The relations can be set via the function
  
Rules can be set through the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetRelations(R);
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetRules(Rules);
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where <tt>R</tt> is a LIST of properly represented relations. One can set the relations to empty via the function
  
where Rules is a properly represented Rules. And Rules can be reset to empty through the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.UnsetRelations();
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetRules();
 
  
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(f) Th following function gives basic information on the working ring.
  
(g) There is a function to get general information about ring environment.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.RingEnv();
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.RingEnv();
 
  
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(g) For most computations, there are three different functions having the same functionality but under different settings. Let us take addition as an example. There are three functions, namely MRAdd(X,Ordering,Relations,F1,F2), Add(F1,F2) and BAdd(F1,F2), doing addition over monoid rings, free monoid rings and free monoid rings over the binary field, respectively. For details about how to use each of them, please check relevant functions.
  
{{ApCoCoAServer}}
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[[Category:ApCoCoA-1 Manual]]
[[Category:ApCoCoA_Manual]]
 

Latest revision as of 15:17, 2 October 2020

The package gbmr contains numbers of functions for basic computations and Groebner basis computations in non-commutative algebras, such as finitely generated free monoid rings (or non-commutative polynomial rings, non-commutative free associative algebras), finitely presented monoid rings, group ring, etc., over the field of rational numbers Q or over finite fields Z/(p) where p is a prime. More precisedly, this package enables us to do computations as addition, subtraction and multiplication of two non-commutative polynomials, getting the leading word and leading coefficient of a non-zero polynomial, computing the normal remainder of a polynomial w.r.t. a list of polynomials, interreducing a lists of polynomials, enumerating (reduced) (partial) Groebner bases of finitely generated two-sided ideals, and computing truncated Groebner basis of a finitely and homogeneously generated two-sided ideals, etc. Consequently, this package can be applied to many algebraic applications, for instance, enumerating a Macaulay's basis and the values of the Hilbert function of a finitely generated K-algbera, computing leading word ideals, intersections of ideals, and kernels of K-algebra homomorphisms, and so on.


Generally speaking, a finitely presented monoid ring is defined by P=K<X|R>, where K is a field, X is a finite alphabet (or a finite set of indeterminates), and R is a finite set of relations. Clearly, we have P=K<X|R> is isomorphic to K<X>/<R>, where K<X> is the free monoid ring generated by X over K and <R> is the two-sided ideal generated by R.


Important issues about this package:

(a) Predefined alias for this package is as follows.

             Alias NCo := $apcocoa/gbmr;

Note that, before ApCoCoA 1.9.0, the alias for this package is NC. However, since ApCoCoA 1.9.0, the alias NC has been used for the ApCoCoA package ncpoly.


(b) By default, the field K is the field of rational numbers. It can be set to a finite field through the functions

             NCo.SetFp(); and NCo.SetFp(P);

where P is a prime number. The former sets the finite field to the binary field {0,1}, and the latter to the finite field {0,1,2,...P-1}. One can reset K to rational numbers via the function

             NCo.UnsetFp();


(c) The alphabet X is represented as a STRING of letters. Every letter in X must have a unique appearance. The order of letters in X is important since it will induce word orderings on the free monoid <X> (see NCo.SetOrdering). The alphabet X is set via the function

             NCo.SetX(X);

where X is a STRING of letters.

       (c.1) Each word (term) in the free monoid <X> is represented as a STRING with all letters coming from X. For example, the word

             w=ba

is represented as

             W:="ba";

      Note that the identity element in <X> is the empty word which is represented as the empty STRING "".

       (c.2) Each non-commutative polynomial is represented as a LIST of monomials, and each monomial is represented as a LIST consisting of an element (coefficient) in K and a word (term) in <X>. For example the polynomial

             f=ab+2b^2+3

is represented as

             F := [[1,"ab"],[2,"bb"],[3,""]];

      Note that the zero polynomial 0 is represented as the empty LIST [].

       (c.3) In the case that K={0,1}, every polynomial can be represented as a LIST of words (terms) in <X>. For example, the polynomial

             p=ab+b^2+1

is represented as

             P := ["ab","bb",""];

Notice that this representation is ONLY applied to computations in free monoid rings over the binary field {0,1}. See functions with the prefix "B" for more details.


(d) A word ordering on a monoid is a well-ordering that is compatible with multiplication. One can set word orderings via the function

             NCo.SetOrdering(Ordering);

where Ordering is a STRING indicating which ordering we are going to work with. Currently, the package only supports the length-lexicographic ordering ("LLEX"), an elimination ordering ("ELIM") and the length-reverse-lexicographic ordering ("LRLEX"). We refer to NCo.SetOrdering for the definitions of these orderings. The default ordering is "LLEX". Note that word orderings are induced by the order of letters in X. For example, X:="abc"; Ordering:="LLEX"; means the length-lexicographic word ordering induced by a>b>c.

(e) For a finitely presented monoid ring P=K<X|R>, the set R of relations is represented as a LIST. and each relation in R is represented as a LIST composed of two words in <X>. For example, the relations

             R={ba=ab, ca=ac, cb=bc}

is represented as

             R:= [["ba","ab"], ["ca","ac"], ["cb","bc"]];

The relations can be set via the function

             NCo.SetRelations(R);

where R is a LIST of properly represented relations. One can set the relations to empty via the function

             NCo.UnsetRelations();


(f) Th following function gives basic information on the working ring.

             NCo.RingEnv();


(g) For most computations, there are three different functions having the same functionality but under different settings. Let us take addition as an example. There are three functions, namely MRAdd(X,Ordering,Relations,F1,F2), Add(F1,F2) and BAdd(F1,F2), doing addition over monoid rings, free monoid rings and free monoid rings over the binary field, respectively. For details about how to use each of them, please check relevant functions.