Some examples of bands with two components. (Q987155)

The paper is motivated by the following open problem: find necessary and sufficient conditions on a band \(S\) (a semigroup consisting of idempotents) in order that \(S\) is a linear semigroup (that is, \(S\) embeds into the multiplicative monoid of \(n\times n\) matrices over a field \(K\), for some \(n\geq 1\)). It is known that every band \(S\) is of the form \(S=\bigcup_{\gamma\in\Gamma}S_\gamma\), a semilattice \(\Gamma\) of rectangular bands \(S_\gamma\) (called components of \(S\)). Moreover, if \(S\) is linear then the semilattice \(\Gamma\) must be finite and every rectangular band is linear. So it is natural to start with the case where \(|\Gamma|=2\). Some progress on the above problem has been made by \textit{F. Cedó} and the reviewer [in Publ. Mat., Barc. 53, No. 1, 119-140 (2009; Zbl 1178.20054)] and by the authors in the paper [in Commun. Algebra 38, No. 11, 4117-4129 (2010; Zbl 1216.20055)]. In the paper under review this work is continued. An example is constructed, showing that certain natural finiteness conditions on one-sided annihilator ideals and on one-sided annihilator congruences in \(S\) are not sufficient for \(S\) to be linear, and a new conjecture is proposed. Moreover, an example of a band \(S\) with two components \(E\), \(F\) satisfying \(EFE=F\) is constructed such that certain commutative algebra \(R\) canonically associated to \(S\), in such a way that the semigroup algebra \(K[S]\) embeds into the algebra of upper triangular matrices \(T_7(R)\), does not satisfy the ascending chain condition on annihilator ideals, whence \(R\) itself does not embed into a matrix ring over a field.