The translational hull of reductive matrix 0-bands (Q1073188)

A binary relation \(\triangleleft\) on a set W is called a weak order if \(a\triangleleft x\) equivalent to \(b\triangleleft x\) for all x implies that \(a=b\), and \(x\triangleleft a\) equivalent to \(x\triangleleft b\) for all x implies that \(a=b\). The main result of this paper is that the semigroup of residuated mappings on a weakly ordered set is, up to isomorphism, the translational hull of a reductive square matrix 0-band.