Remarks on regular congruences on po-semigroups (Q5929677)

The author defines congruences on a po-semigroup \(S\). He also defines semilattice congruences, complete semilattice congruences and the least complete semilattice congruences on \(S\). Finally, a regular congruence \(\rho\) on \(S\) and the regular order on \(S/\rho\) with respect to \(\rho\) are defined. The regular order on \(S/\rho\) with respect to a regular congruence on \(S\) is in general not unique. The following theorem is established: The least regular order on \(S/\rho\) with respect to a regular congruence \(\rho\) on \(S\) exists. It is also shown that not all congruences on a po-semigroup are regular.