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Package gbmr is designed to enable us to do basic computations, such as addition(Add(F1,F2)), subtraction(Subtract(F1,F2)) and multiplication(Multiply(F1,F2)), leading term(LT(F)), leading coefficient(LC(F)), normal remainder(NR(F,G)), (partial) Groebner basis(GB(G)), reduced (partial) Groebner basis (ReducedGB(G)) and so on, over non-commutative algebra, i.e. finitely presented monoid ring, free associative algbra, and group ring. As a consequence, the package can also be applied to many algebraic applications, for instance leading term ideal(LTIdeal(G)), Hilbert function(HF(Gb)), etc.
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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 each computation mentioned above, there are 3 different functions having the same functionality but under different settings. Take addition for an example, there are MRAdd(X, Ordering, Relations, F1, F2), Add(F1, F2) and GAdd(F1, F2) doing addition over monoid ring, free associative algebra and group ring, respectively. For details about how to use each of them, please check relevant functions.
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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>.
  
  
Generally speaking, 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 (a finite set of letters or indeterminates), 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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<strong>Important issues about this package:</strong>
  
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(a) Predefined alias for this package is as follows.
  
Things to know about this package.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Alias NCo := $apcocoa/gbmr;
  
(a) Predefined alias for this package is
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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; Alias NC := $apcocoa/gbmr;
 
  
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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
  
(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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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetFp(); and NCo.SetFp(P);
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetFp(); and NC.SetFp(Prime);
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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
  
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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&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.UnsetFp();
 
  
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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
  
(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.SetX(X);
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetX(X);
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where <tt>X</tt> is a STRING of letters.
  
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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&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
  
&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; <tt>w=ba</tt>
  
However I fail to find a proper situation to use it currently.
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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";
  
(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;Note that the identity element in <tt><X></tt> is the empty word which is represented as the empty STRING "".
  
For example, X:="abc"; Ordering:="ELIM"; means elimination ordering induced from <tt>a>b>c</tt>.  
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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
  
Ordering can be set through the function
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>f=ab+2b^2+3</tt>
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetOrdering(Ordering);
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is represented as
  
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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; F := [[1,"ab"],[2,"bb"],[3,""]];
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetOrdering();
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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
  
(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; <tt>p=ab+b^2+1</tt>  
  
&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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is represented as  
  
&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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; P := ["ab","bb",""];
  
&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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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.
  
For example, X := "abc"; Relations := [["ba","ab"], ["ca","ac"], ["cb","bc"]]; means Relations generated by <tt>{ba=ab, ca=ac, cb=bc}</tt>.
 
  
Relations can be set through the function
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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
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetRelations(Relations);
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetOrdering(Ordering);
  
where Relations is a properly represented Relations. And Relations can be reset to empty through the function
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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>.
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetRelations();
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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
  
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; <tt>R={ba=ab, ca=ac, cb=bc}</tt>
  
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is represented as
  
(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></tt> (or <tt>K<X|R></tt>).
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; R:= [["ba","ab"], ["ca","ac"], ["cb","bc"]];
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(f0) Each polynomial in <tt>K<X></tt> (or <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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The relations can be set via the function
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;For example, X := "abc"; P := [[1,"ab"], [1,""]]; means <tt>P=ab+1</tt>.  
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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;Note that 0 polynomial is represented as an empty LIST [].
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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
  
For example, X := "ab"; Rules := [["ba",  [[1,"ab"], [1,""]]]]; means Rules generated by <tt>{ba=ab+1}</tt>.  
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.UnsetRelations();
  
Rules can be set through the function
 
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.SetRules(Rules);
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(f) Th following function gives basic information on the working ring.
  
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.RingEnv();
  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.UnsetRules();
 
  
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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.
  
(g) There is a function to get general information about ring environment.
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[[Category:ApCoCoA-1 Manual]]
 
 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NC.RingEnv();
 
 
 
 
 
{{ApCoCoAServer}}
 
[[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.