On associative superalgebras of matrices. (Q1880821)

Let \(G\) be a group, a \(G\)-grading of a matrix ring is called good if the matrix units \(e_{ij}\) are homogeneous. All good gradings of the full matrix ring \(M_k(F)\) were classified by \textit{S. Dăscălescu, B. Ion, C. Năstăsescu}, and \textit{J. Ríos Montes} [J. Algebra 220, No. 2, 709-728 (1999; Zbl 0947.16028)]. Now the authors consider gradings by semigroups. It is well-known that there exist five isomorphism types of semigroups with two elements. The authors classify all gradings for the full matrix algebra \(M_2(F)\) by these semigroups different from \(\mathbb{Z}_2\). It is proved that all these gradings are good. If \(F\) is not algebraically closed, the above reference yields examples of \(\mathbb{Z}_2\)-gradings for \(M_2(F)\) which are not isomorphic to good ones. The authors also classify all gradings by the semigroups with two elements for the algebra of \(2\times 2\) upper triangular matrices. The existence of non-good gradings is also considered. In some cases it is assumed that the characteristic of the base field \(F\) is different from 2.