English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study a question which can be roughly stated as follows: Given a (unitalor nonunital) algebra $A$ together with a Gr\"obner-Shirshov basis $G$,consider the free operated algebra $B$ over $A$, such that the operatorsatisfies some polynomial identities $\Phi$ which are Gr\"obner-Shirshov in thesense of Guo et al., when doesthe union $\Phi\cup G$ will be an operatedGr\"obner-Shirshov basis for $B$? We answer this question in the affirmativeunder a mild condition in our previous work with Wang. When this condition issatisfied, $\Phi\cup G$ is an operated Gr\"obner-Shirshov basis for $ B$ and asa consequence, we also get a linear basis of $B$. However, the condition couldnot be applied directly to differential type algebras introduced by Guo, Sitand Zhang, including usual differential algebras.  This paper solves completely this problem for differential type algebras.Somenew monomial orders are introduced which, together with some known ones, permitthe application of the previous result to most of differential type algebras,thus providing new operated GS bases and linear bases for these differentialtype algebras.Versions are presented both for unital and nonunital algebras.However, a class of examples are also presented, for which the naturalexpectation in the question is wrong and these examples are dealt with bydirect inspection.

Gr\"obner-Shirshov bases and linear bases for free differential type algebras over algebras