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Package gbmr is designed to provide basic operations over monoid rings and compute Groebner bases of finite generated ideals.
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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.
  
For the field of rationals '''Q''' and a monoid '''M''' presented by a string rewriting system ('''Alphabet''', '''Rs'''), where '''Alphabet''' is finite alphabet and '''Rs''' is set of '''Relation'''s, let Q[M] denote the ring of all finite formal sums (called '''Polynomial'''s) a_{1}*w_{1}+ a_{2}*w_{2} +...+a_{n}*w_{n} with coefficients a_{i} in Q\{0} and terms w_{i} in M. This ring is called the '''monoid ring''' of M over Q.
 
  
Notice that in this package
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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>.
  
(i) '''Alphabet''' is of '''STRING''' type. Every letters in Alphabet MUST appear ONLY ONCE, since the order of letters in '''Alphabet''' will induce the order of terms later. For example, Alphabet:="abc"; Order:="LLEX"; means length-lex ordering induced by the ordering a>b>c.
 
  
(ii) Each '''Polynomial''' is represented as a LIST of LISTs, which are pairs of '''Term''' (of '''STRING''' type) and corresponding '''Coefficient'''. '''Polynomial:=[[Term,Coefficient],...,[Term, Coefficient]]'''. For example, polynomial F:=a+1 is represented as F:=[[1,"a"], [1,""]].
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<strong>Important issues about this package:</strong>
  
(iii) Each '''Relation''' is a pair '''[L:Term, R:Polynomial]''', where '''L''' is left side of relation and '''R''' is right side of relation. For example, R:=["ba",[[1,"ab"], [1,""]]]; means relation ba->ab+1.  
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(a) Predefined alias for this package is as follows.
  
(iv) '''Order''' is a total well-founded admissible ordering induced by the order of letters in '''Alphabet'''. It is represented as a '''STRING''', which is the name of ordering. And "LLEX" (length-lex ordering) is the only ordering supported currently.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Alias NCo := $apcocoa/gbmr;
  
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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.
  
Let p, f be two non-zero polynomials in Q[M]. We say f '''prefix reduce'''s p to q at a monomial a*t of p in one step, denoted by p-->_{f}q if
 
  
  (1) LT(f)w = t for some w in M, i.e., LT(f) is a prefix of t, and
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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
  (2) q = p-a*LT(f)^{-1}*f*w.
 
  
A set G is said to be a '''Groebner basis''' with respect to the reduction -->, if <-->_{G} = Equiv_{Ideal(G)} and -->_{G} is confluent.
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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
  
{{ApCoCoAServer}}
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.UnsetFp();
[[Category:ApCoCoA_Manual]]
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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
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetX(X);
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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
 +
 
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>w=ba</tt>
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is represented as
 +
 
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; W:="ba";
 +
 
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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 "".
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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
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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
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  F := [[1,"ab"],[2,"bb"],[3,""]];
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Note that the zero polynomial <tt>0</tt> is represented as the empty LIST [].
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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
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>p=ab+b^2+1</tt>
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is represented as
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; P := ["ab","bb",""];
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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.
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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
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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>.
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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
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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>
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is represented as
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; R:= [["ba","ab"], ["ca","ac"], ["cb","bc"]];
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The relations can be set via the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetRelations(R);
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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
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.UnsetRelations();
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(f) Th following function gives basic information on the working ring.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.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.
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[[Category:ApCoCoA-1 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.