Singular Manual: Monomial orderings on free algebras

| ### 7.9.2 Monomial orderings on free algebras Monomial orderings are of critical importance when dealing with Groebner bases. Especially in the non-commutative Groebner theory up to a degree (length) bound it might be vital to have many choices for orderings since it is not guaranteed that there exists a finite Groebner bases with respect to the given ordering. Let = { ,..., } be a set of symbols. A total ordering < on the free monoid with as the neutral element is called a monomial ordering if it is a well-ordering, i.e., every non empty subset has a least element with respect to <, and it respects multiplication, that is implies for all , , and in . Note that the latter implies for all in . The left lexicographical ordering on with ... is defined as follows: For arbitrary , in we say that , if or and holds. Note: left lex is not a monomial ordering, though it is a natural choice to break ties in further definitions. In a similar manner one can define the right lexicographical ordering. On the monoid define the weight homomorphism , uniquely determined by in for . As a special case, define the length len: by for . For any ordering << on and any weight define an ordering <, the -weight extension of << as follows: For arbitrary , in we say that if or and << holds. An ordering < on eliminates a certain subset if for all one has . In a ring declaration, LETTERPLACE supports the following monomial orderings. We illustrate each of the available choices by an example on the free monoid , , , where we order the monomials , , , , , , , , , and correspondingly. ```dp`'` The degree right lexicographical ordering is the length-weight extension of the right lexicographical ordering. With respect to the ordering ``dp'`, the test monomials are ordered as follows: ```Dp`'` The degree left lexicographical ordering is the length-weight extension of the left lexicographical ordering. With respect to the ordering ``Dp'`, the test monomials are ordered as follows: ```Wp(w)` for intvec w'` The weighted degree left lexicographical ordering is the -weight extension of the left lexicographical ordering with weight uniquely determined by strict positive . With respect to the ordering ``Wp(1, 2, 1)'`, the test monomials are ordered as follows: ```lp`'` Let be weights uniquely determined by for where denotes the Kronecker delta. Let be the -weight extension of the left lexicographical ordering on and inductively be the -weight extension of for all . The monomial ordering lp corresponds to and eliminates for all . The monomial ordering ``lp'` corresponds to and eliminates { ,..., } for all <= < . With respect to the ordering ``lp'`, the test monomials are ordered as follows: ```rp`'` Let be weights uniquely determined by for where denotes the Kronecker delta. Let be the -weight extension of the left lexicographical ordering on and inductively be the -weight extension of for all . The monomial ordering `rp` corresponds to and eliminates for all . The monomial ordering ``rp'` corresponds to and eliminates { ,..., } for all . With respect to the ordering ``rp'`, the test monomials are ordered as follows: ```(a(v), ordering)` for intvec v'` For weight determined by with and monomial ordering on , the -weight extension of corresponds to `(a(v), o)`. As a choice for there are currently two options implemented, which are `dp` and `Dp`. Notice that this ordering eliminates . With respect to the ordering ``( a(1, 0, 0), Dp)'`, the test monomials are ordered as follows: With ordering ``( a(1, 1, 0), Dp)'` one obtains: The examples are generated by the following code but with customized orderings denoted above.
    LIB "freegb.lib";
    ring r = 0, (x1,x2,x3),Dp; // variate ordering here
    def R = freeAlgebra(r, 4);
    setring R;
    poly wr = x1*x1*x1+x3*x3*x3+x1*x2*x3+x3*x2*x1+x2*x2+x2*x3+x1*x3+x3*x1+x1+x2+x3;
    wr;
    ==> x1*x1*x1+x1*x2*x3+x3*x2*x1+x3*x3*x3+x1*x3+x2*x2+x2*x3+x3*x1+x1+x2+x3

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