|Title||Gr\"obner bases on algebras based on well-ordered semigroups.|
|Author(s)|| Yuji Kobayashi|
|Type||Book, Chapter in Book, Conference Proceeding|
|Abstract||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.
|Keywords||Gröbner basis; well-ordered semigroup; rewriting system; critical pair; z-element |
|Publisher||Hackensack, NJ: World Scientific|