Difference between revisions of "Category:ApCoCoA-1:Package gbmr"

Package gbmr is designed to provide basic operations (addition, subtraction, multiplication) over monoid rings and Groebner basis computations for finite generated (one and two-sided) ideals.

Let Q be rational field and M=<X, R> be a finited presented monoid, where X is a finite alphabet and R is a finite set of relations. A monoid ring of M over Q, denoted by Q[M], is a ing of all finite formal sums (called polynomials) 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.

Note that

(i) X is of STRING type in this package. Every letters in X MUST appear only once. The order of letters in X is very important, since it induces a term ordering later. For example, X:="abc"; Order:="LLEX"; means a length-lexicographic ordering induced by a>b>c.

(ii) Each element (relation) in R is of form [L, R], where L and R are terms in M. Each term in M is represented as a STRING. For example, xy^2x is represented as "xyyx", and relation (yx, xy) is represented as ["yx", "xy"].

(iii) Each polynomial in Q[M] is represented as a LIST of LISTs, which are pairs of form [a_{i}, w_{i}]. For example, polynomial F:=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].

(iv) Ordering is of STRING type, which is an abbreviated name of a term ordering. For exapme, "LLEX" stands for a length-lexicographic ordering and "ELIM" stands for an elimination ordering. These two term orderings are the only supported orderings currently.

Let p, f be two non-zero polynomials in Q[M]. We say f prefix reduces 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
 (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.

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.