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6.1.9 Package gbmr
The following are commands and functions within the package gbmr:
NC.Add Addition of two polynomials in a free monoid ring.
NC.Deg (Standard) degree of a polynomial in a free monoid ring.
NC.FindPolynomials Find polynomials with specified alphabet (set of indeterminates) from a list of polynomials in monoid rings.
NC.GAdd Addition of two polynomials in a free group ring over F2.
NC.GB Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring (using the Buchberger procedure).

Given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). Note that it may not exist finite Groebner bases of I w.r.t. Ordering.
NC.GDeg (Standard) degree of a polynomial in a free monoid ring over F2.
NC.GGB Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring over F2 (using Buchberger procedure).

Given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). Note that it may not exist finite Groebner bases of I w.r.t. Ordering.
NC.GHF Compute the values of the Hilbert function of a finitely generated F2-algebra.

For every i in N, we let F_{i} be the F2-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of F2. Let I be an ideal of F2. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of F2/I. The Hilbert function of F2-algebra F2/I is a map from N to N defined by mapping i to dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I)).
NC.GInterreduction Interreduce a list (set) of polynomials in a free monoid ring over F2.

Note that, given an admissible ordering Ordering, a set of non-zero polynomial G is called interreduced w.r.t. Ordering if no element of Supp(g) is contained in LT(G\{g}) for all g in G.
NC.GIsGB Check whether a finite list (set) of non-zero polynomials in a free monoid ring over F2 is a Groebner basis.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). The function check whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if all the S-polynomials of obstructions have the zero normal remainder w.r.t. G.
NC.GLC Leading coefficient of a polynomial in a free monoid ring over F2.
NC.GLT Leading term of a polynomial in a free monoid ring over F2.
NC.GMB Macauley basis of an F2-algebra.
NC.GMultiply Multiplication of two polynomials in a free monoid ring over F2.
NC.GNR Normal remainder of polynomial with respect to a list of polynomials in a free monoid ring over F2.
NC.GReducedGB Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring over F2.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). A Groebner basis w.r.t. Ordering is called a reduced Groebner basis w.r.t. Ordering if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. Ordering. However, it is not necessarily finite.
NC.GSubtract Subtraction of two polynomials in a free monoid ring over F2.
NC.GTruncatedGB Compute a degree truncated Groebner basis of a finitely generated homogenous two-sided ideal in a free monoid ring over F2.
NC.HF Compute the values of the Hilbert function of a finitely generated K-algebra.

For every i in N, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of K. Let I be an ideal of K. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of K/I. The Hilbert function of K-algebra K/I is a map from N to N defined by mapping i to dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I)).
NC.Interreduction Interreduce a list (set) of polynomials in a free monoid ring.

Note that, given an admissible ordering Ordering, a set of non-zero polynomial G is called interreduced w.r.t. Ordering if no element of Supp(g) is contained in LT(G\{g}) for all g in G.
NC.Intersection Enumerate a (partial) Groebner basis of the intersection of two finitely generated two-sided ideals in a free monoid ring.
NC.IsFinite Check whether the K-dimension dim(K/) is finite, where is an monoid ideal generated by a finite set of monoid M.
NC.IsGB Check whether a finite list (set) of non-zero polynomials in a free monoid ring is a Groebner basis.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). The function check whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if all the S-polynomials of obstructions have the zero normal remainder w.r.t. G.
NC.IsHomog Check whether a polynomial of a list of polynomials is homogeneous in a free monoid ring.
NC.KernelOfHomomorphism Enumerate a (partial) Groebner basis of the kernel of a K-algebra homomorphism.
NC.LC Leading coefficient of a polynomial in a free monoid ring.
NC.LT Leading term of a polynomial in a free monoid ring.
NC.LTIdeal Enumerate a (partial) (two-sided) leading-term ideal of a finitely generated ideal in a free monoid ring.
NC.MB Enumerate Macaulay basis of a K-algebra.
NC.MinimalPolynomial Compute a minimal polynomial of an element over the quotient of a free monoid ring if it exists.
NC.MRAdd Addition of two polynomials in a finitely presented monoid ring.
NC.MRDeg (Standard) degree of a polynomial in a finitely presented monoid ring.
NC.MRGB Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal in a finitely presented monoid ring (by the Buchberger procedure).

Given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term set LT{I} (w.r.t. Ordering) as a monoid ideal. Also note that it may not exist finite Groebner bases of I w.r.t. Ordering.
NC.MRHF Hilbert function of a K-algebra.

For every i in N, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of K. Let I be an ideal of K. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of K/I. The Hilbert function of K-algebra K/I is a map from N to N defined by mapping i to dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I)).
NC.MRInterreduction Interreduce a list (set) of polynomials in a finitely presented monoid ring.

Given an admissible ordering Ordering, a set of non-zero polynomial G is called interreduced w.r.t. Ordering if no element of Supp(g) is contained in LT(G\{g}) for all g in G.
NC.MRIntersection Enumerate a (partial) Groebner basis of the intersection of two finitely generated two-sided ideals in a finitely presented monoid ring.
NC.MRIsGB Check whether a finite list (set) of non-zero polynomials in a finitely presented monoid ring is a Groebner basis.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term set LT{I} (w.r.t. Ordering) as a monoid ideal. The function check whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if all the S-polynomials of obstructions have the zero normal remainder w.r.t. G.
NC.MRIsHomog Checks whether a polynomial or a set (LIST) of polynomials is homogeneous in a finitely presented monoid ring.
NC.MRKernelOfHomomorphism Enumerate a (partial) Groebner basis of the kernel of a K-algebra homomorphism.
NC.MRLC Leading coefficient of a polynomial in a finitely presented monoid ring.
NC.MRLT Leading term of a polynomial in a finitely presented monoid ring.
NC.MRLTIdeal Enumerate a (partial) (two-sided) leading-term ideal of a finitely generated ideal in a finitely presented monoid ring.
NC.MRMB Enumerate Macaulay basis of a K-algebra.
NC.MRMinimalPolynomial Compute a minimal polynomial of an element over the quotient of a finitely presented monoid ring if it exists.
NC.MRMultiply Multiplication of two polynomials in a finitely presented monoid ring.
NC.MRNR Normal remainder of a polynomial with respect to a list of polynomials in a finitely presented monoid ring.
NC.MRReducedGB Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a finitely presented monoid ring.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term set LT{I} (w.r.t. Ordering) as a monoid ideal. A Groebner basis w.r.t. Ordering is called a reduced Groebner basis w.r.t. Ordering if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. Ordering. However, it is not necessarily finite.
NC.MRSubtract Subtraction of two polynomials in a finitely presented monoid ring.
NC.Multiply Multiplication of two polynomials in a free monoid ring.
NC.NR Normal remainder of polynomial with respect to a list of polynomials in a free monoid ring.
NC.PrefixGB Prefix Groebner basis of a finitely generated (right) ideal in a finitely presented monoid ring.
NC.PrefixInterreduction Prefix interreduction of a list of polynomials in a finitely presented monoid ring.
NC.PrefixNR Prefix normal remainder of a polynomial with respect to a list of polynomials in a finitely presented monoid ring.
NC.PrefixReducedGB Prefix reduced Groebner basis of a finitely generated (right) ideal in a finitely presented monoid ring.
NC.PrefixSaturation Prefix saturation of a polynomial in a finitely presented monoid ring.
NC.ReducedGB Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a free monoid ring.

Note that, given an ideal I and an admissible ordering Ordering, a set of non-zero polynomials Gb is called a Groebner basis of I w.r.t. Ordering if the leading term set LT{Gb} (w.r.t. Ordering) generates the leading term ideal LT(I) (w.r.t. Ordering). A Groebner basis w.r.t. Ordering is called a reduced Groebner basis w.r.t. Ordering if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. Ordering. However, it is not necessarily finite.
NC.SetFp Set coefficient field to a finite field.
NC.SetOrdering Set an admissible ordering on .
NC.SetRelations Set relations for a finitely presented monoid ring.
NC.SetRules Set the (rewriting) rules over monoid rings.
NC.SetX Set alphabet (set of indeterminates).
NC.Subtract Subtraction of two polynomials in a free monoid ring.
NC.TruncatedGB Compute a degree truncated Groebner basis of a finitely generated homogenous two-sided ideal in a free monoid ring.
NC.UnsetFp Set coefficient field to the default coefficient field Q, i.e. RAT in CoCoAL.
NC.UnsetOrdering Set the current admissible ordering to the default ordering LLEX (length-lexicographic ordering).
NC.UnsetRelations Set relations to the empty set.
NC.UnsetRules Set rewriting rules to the empty set.
NC.UnsetX Set alpbabet (set of inderminates) to the empty string. (It might be a useless function.)




For details look up each item by name. Online, try ?ItemName or H.Syntax("ItemName").