On the structure of semigroups of idempotent matrices. (Q996233)

Bands (idempotent semigroups) of matrices are considered. A band \(S\) is called regular if it satisfies the identity \(axaya=axya\). A band \(S\) is called pure if \(S/\mathcal D\) is a chain (\(\mathcal D\) is the known Green's relation). It is proved that any pure regular band \(S\) of matrices admits a simultaneous LU decomposition, i.e., it exists a basis such that all matrices in \(S\) are in the standard form and each matrix is a product of a lower diagonal matrix by an upper diagonal matrix. In the case when the band \(S\) of matrices forms a double band called a skew lattice the standard form can be obtained without the assumption of purity.