A band is a semigroup consisting of idempotents. It is proved that for any field $K$ and any band $S$ with finitely many components, the semigroup algebra $K[S]$ can be embedded in upper triangular matrices over a commutative $K$-algebra. The proof of a theorem of Malcev on embeddability of algebras into matrix algebras over a field is corrected and it is proved that if $S=F\cup E$ is a band with two components $E$, $F$ such that $F$ is an ideal of $S$ and $E$ is finite, then $S$ is a linear semigroup. Certain sufficient conditions for linearity of a band $S$, expressed in terms of annihilators associated to $S$, are also obtained.
F. Cedó. J. Okniński. "Faithful linear representations of bands." Publ. Mat. 53 (1) 119 - 140, 2009.