On regular, strongly regular congruences on ordered semigroups
If \((S,\cdot,\leq)\) is a partially ordered (p.o.) semigroup and \(\rho\) is a congruence on \((S,\cdot)\), then, in general, the quotient semigroup \((S/\rho,*)\) does not inherit from \((S,\leq)\) a nontrivial partial order \(\preceq\) such that \((S/\rho,*,\preceq)\) forms again a p.o. semigroup. In this paper, \(\rho\) is called regular if there exists a partial order \(\preceq\) on \(S/\rho\) such that \((S/\rho,*,\preceq)\) forms a p.o. semigroup and if the natual homomorphism of \(\rho\) is isotone. First, it is shown that for every ideal \(I\) of \(S\) (defined as a semigroup ideal of \((S,\cdot)\) which is also an order ideal of \((S,\leq))\), the Rees congruence of \(I\) is regular. Using a result on pseudo-orders due to \textit{N. Kahayopulu} and \textit{M. Tsingelis} [Semigroup Forum 50, 389-392 (1995; Zbl 0828.06010)], it is shown that a congruence \(\rho\) on a p.o. semigroup \(S\) is regular if and only if there exists a p.o. semigroup \(T\) and a homomorphism \(\varphi:S\to T\) such that \(a\rho b\Leftrightarrow\varphi(a)=\varphi(b)\). A congruence \(\rho\) on \((S,\cdot,\leq)\) is called strongly regular if \(\rho\circ \leq\) is contained in \(\leq \circ\rho\). Characterizations of regular and strongly regular congruences are given, in particular in the case that \((S,\cdot,\leq)\) is a lattice-ordered semigroup. Finally, it is shown that the p.o. set of all (strongly) regular congruences forms a complete lattice, which is not a sublattice of the congruence lattice of \((S,\cdot)\) in general.
scientific article; zbMATH DE number 1523487