Faithful linear representations of bands. (Q998729)

An attractive problem in ring theory is the embeddability of a ring \(R\) into the full or triangular matrix ring over a commutative ring. In this paper the authors investigate such questions for semigroup algebras \(K[S]\) of a band \(S\) and a field \(K\). Recall that a band is a semigroup consisting of idempotents. Such a semigroup is a semilattice \(S=\bigcup_{\alpha\in\Gamma}S_\alpha\) of rectangular bands \(S_\alpha\). These are called the components of \(S\). In the first part of the paper it is proved that if a band \(S\) has finitely many components then \(K[S]\) can be embedded in upper triangular matrices over a commutative \(K\)-algebra. An explicit embedding is given in case \(S\) has two components. In the second part, the proof of a theorem of \textit{A. I. Mal'tsev} [Mat. Sb., N. Ser. 13(55), 263-286 (1943; Zbl 0060.07804), Theorem 10] on embeddability of algebras into matrix algebras over a field is corrected. Furthermore, it is proved that if \(S=F\cup E\) is a band with two components \(E\) and \(F\), such that \(F\) is an ideal of \(S\) and \(E\) is finite, then \(S\) is a linear semigroup. For some special classes of bands satisfying finiteness conditions on annihilators, some explicit embeddings of the respective semigroups, and their semigroup algebras, into triangular matrices over fields are given.