Balanced order relations on completely simple semigroups (Q799709)

It is well known that the congruences of a semisimple semigroup S given in its Rees representation M(G;I,\(\Lambda\) ;P) can be reduced to triples consisting of a normal subgroup of G and a pair of equivalence relations on the sets I and \(\Lambda\) respectively [see \textit{J. M. Howie}, An introduction to semigroup theory (1976; Zbl 0355.20056), Theorem 4.23]. The author looks for a similar theorem for partially ordered completely simple semigroups S and he succeeds for balanced partial orders on S, i.e. \((i,j)\in\zeta_ I\), where \(\zeta_ I\) is the projection of \(\zeta\) on I, implies that the idempotents \((p^{-1}_{\lambda i};i,\lambda)\) and \((p^{-1}_{\lambda j};j,\lambda)\) are \(\zeta\) - related for all \(\lambda \in\Lambda \), and similarly for the other index set \(\Lambda\). He proves that the balanced partial orders on S are in order-preserving bijection onto the set of all admissible triples (Q;\(\xi\),\(\eta)\) where Q is the positive cone of a partial order of the structure group G of S and the partial orders \(\xi\) and \(\eta\) of I and \(\Lambda\) respectively satisfy: (i,j)\(\in\xi \) implies \(p^{-1}_{\mu i}p_{\mu j}p_{\lambda j}p_{\lambda i}\in Q\) for all \(\lambda\),\(\mu \in\Lambda \), and similarly for \(\eta\).