Category:ApCoCoA-1:Package gbmr

Package gbmr is designed to enable us to do basic computations, such as addition (via Add(F1,F2)), subtraction (via Subtract(F1,F2)) and multiplication (via Multiply(F1,F2)), leading term (via LT(F)), leading coefficient(via LC(F)), normal remainder (via NR(F,G)), (partial) Groebner basis(via GB(G)), reduced (partial) Groebner basis (via ReducedGB(G)) and so on, over non-commutative algebras, i.e. over finitely generated free monoid rings (or free associative K-algebras or non-commutative polynomial rings), finitely presented monoid rings, group ring, etc. Consequently, the package can be applied to many algebraic applications, for instance the computations of leading term ideal (via LTIdeal(G)), kernel of K-algebra homomorphism (via KernelOfHomomorphism(X1,X2,Images)), Hilbert function (via HF(Gb)), etc.

For each computation mentioned above, there are three different functions having the same functionality but under different settings. Let us take addition for an example. There are three functions, namely 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.

Generally speaking, a finitely presented monoid ring is defined by P=K<X|R>=K<X>/<R>, where K is a field, X is a finite alphabet (a finite set of indeterminates), and R is a finite set of relations. Particularly, P becomes a free associative K-algebra if R is empty.

Things to know about this package.

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

             Alias NC := $apcocoa/gbmr;

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

             NC.SetFp(); and NC.SetFp(Prime);

where Prime is a prime number. The prevouse one sets finite field to F2=Z/(2) and the latter to F_{Prime}=Z/(Prime). One can reset K to rational number via the function

             NC.UnsetFp();

(c) The alphabet X is represented as a STRING of letters. Every letter in X should have a unique occurrence. The order of letters in X is important since it induces an admissible ordering specified by Ordering. The alphabet X is set through the function

             NC.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, X := "abc"; W := "ba"; means a word W=ba.

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

      (c.2) Each polynomial in K<X> (or K<X|R>) 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, X := "abc"; P := [[1,"ab"],[1,"bb"],[1,""]]; means a polynomial P=ab+b^2+1.

      Note that the zero polynomial is represented as an empty LIST []. Also note that, in the case that K=F2, every polynomial can be represented as a LIST of words (terms) in <X>. For example, polynomial P=ab+b^2+1 in F_{2}<a,b> is represented as P := ["ab","bb",""]. We refer to the examples of functions in free group rings (functions with the prefix "G") for more details.

(d) An ordering Ordering is a STRING indicating which admissible ordering we are working with. Currently, the package only supports length-lexicographic ordering ("LLEX"), elimination ordering ("ELIM") and degree reverse lexicographic ordering ("DEGREVLEX"), which are induced from the order of letters in X. Note that the default ordering is "LLEX".

For example, X:="abc"; Ordering:="ELIM"; means the elimination ordering induced by a>b>c.

The ordering Ordering is set through the function

             NC.SetOrdering(Ordering);

where Ordering is one of the orderings supported by the package. One can reset Ordering to "LLEX" via the function

             NC.UnsetOrdering();

(e) A set Relations of relations is a finite set represented as a LIST. Each relation in Relations is represented as a LIST composed of two words in <X>.

For example, X := "abc"; Relations := [["ba","ab"], ["ca","ac"], ["cb","bc"]]; means Relations containing ba=ab, ca=ac and cb=bc.

A set of relations is set through the function

             NC.SetRelations(Relations);

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

             NC.UnsetRelations();

Note that using the functions UnsetRelations() is a tricky way to change a monoid ring to a free monoid ring.

(f) A set Rules of rules is a finite set represented as a LIST. Each (rewriting) rule in Rules is represented as a LIST composed of a word in <X> and a polynomial in K<X> (or K<X|R>).

For example, X := "ab"; Rules := [["ba", [[1,"ab"], [1,""]]]]; means Rules containing ba=ab+1.

A set of rules is set through the function

             NC.SetRules(Rules);

where Rules is a set of properly represented rules. One can set Rules to empty via the function

             NC.UnsetRules();

(g) There is a function for showing general information on ring environment.

             NC.RingEnv();