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 set of letters 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. 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) R is of LIST type. Each element in R has form [w_{l}, w_{r}], where w_{l} and w_{r} are terms in M. Each term in M is represented as a STRING. For example, xy^2x is represented as "xyyx", identity element is represented as a empty STRING "", 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,""]]. Zero polynomial is represented as an empty LIST [].

(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 orderings supported by the package currently.

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.