GRÖBNER BASES ON ALGEBRAS BASED ON WELL-ORDERED SEMIGROUPS
This work was partially supported by Grant-in-Aid for Scientific Research (No. 21540048).
We develop the theory of Gröbner bases on an algebra based on a well-ordered semigroup inspired by the discussions in Farkas et al.,3,4 where the authors study multiplicative bases in an axiomatic way. We consider a reflexive semigroup with 0 equipped with a suitable well-order, and use it as a base of an algebra over a commutative ring, on which we develop a Gröbner basis theory.
Our framework is considered to be fairly general and unifies the existing Gröbner basis theories on several types of algebras (ref. 1,6–10). We discuss a Gröbner basis theory from a view point of rewriting systems. We study behaviors of critical pairs in our situation and give a so-called critical pair theorem. We need to consider z-elements as well as usual critical pairs come from overlapping applications of rules.
