Narrow Results

We define the product of admissible abstract kernels of the form $\mathrm{\Phi }:M\to \frac{\mathrm{\text{End}}(G)}{\mathrm{\text{Inn}}(G)}$, where $M$ is a monoid, $G$ is a group and $\mathrm{\Phi }$ is a monoid homomorphism. Identifying $C$-equivalent abstract kernels, where $C$ is the center of $G$, we obtain that the set ${ℳ}(M,C)$ of $C$-equivalence classes of admissible abstract kernels inducing the same action of $M$ on $C$ is a commutative monoid. Considering the submonoid ${ℒ}(M,C)$ of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid ${𝒜}(M,C)=\frac{{ℳ}(M,C)}{{ℒ}(M,C)}$ is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group $H^3(M,C)$.

In this survey article, we report some new results of Gröbner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers.