Enumeration of finite monomial orderings and combinatorics of universal Gröbner bases

The goal of this work is to analyze various classes of finite and total monomial orderings. The concept of monomial ordering plays the key role in the theory of Grobner bases: every basis is determined by a certain ordering. At the same time, in order to define a Grobner basis, it is not necessary to know ordering of all monomials. Instead, it is sufficient to know only a finite interval of the given ordering. We consider combinatorics of finite monomial orderings and its relationship with cells of a universal Grobner basis. For every considered class of orderings (weakly admissible, convex, and admissible), an algorithm for enumerating finite orderings is discussed and combinatorial integer sequences are obtained. An algorithm for computing all minimal finite orderings for an arbitrary Grobner basis that completely determine this basis is presented. The paper presents also an algorithm for computing an extended universal Grobner basis for an arbitrary zero-dimensional ideal.