Decompositions and pseudo-orders of ordered semigroups (Q1434112)

Let \((S,\cdot,\leq)\) be a partially ordered semigroup. A semilattice congruence \(\eta\) on \((S,\cdot)\) is called ``natural ordered'' if \(a\leq b\) \((a,b\in S)\) implies \(ab \eta a\). Following \textit{N. Kahayopulu} and \textit{M. Tsingelis} [Semigroup Forum 50, 392--398 (1995; Zbl 0828.06010)] a binary relation \(\omega\) on \(S\) is called ``pseudo-order'' if \(\omega\) is transitive, compatible with multiplication, and \(a\leq b\Rightarrow a\omega b\). For any pseudo-order \(\omega\), \(\overline\omega= \omega\cap \omega^{-1}\) is a congruence on \((S,\cdot)\) such that \((S/\overline\omega,*, \preccurlyeq)\) is a partially ordered semigroup with respect to \(a\overline\omega\preccurlyeq b\overline\omega\) iff \(x\omega y\) for some \(x\in a\overline\omega\), \(y\in b\overline\omega\). A pseudo-order \(\omega\) on \(S\) is called a ``natural ordered semilattice pseudo-order (nosp)'' if \(\omega\) is a natural ordered semilattice congruence on \(S\). In such a case it follows that \(S\) is decomposed by \(\overline\omega\) into subsemigroups which form a semilattice in the above sense. If this decomposition is ``strong'', \((S,\cdot,\leq)\) is a subdirect product of partially ordered semigroups with minimum zero elements possibly adjoined. The main result of the paper generalizes the description of an arbitrary semilattice congruence on a semigroup by means of sets of filters. It is shown that a pseudo-order \(\omega\) is a nosp on \((S,\cdot,\leq)\) iff \(\omega= \bigcap\omega_F\) with \(F\in{\mathcal F}\), where \({\mathcal F}\) is a set of proper ``filters'' and \(\omega_F= (S\setminus F\times S\setminus F)\cup(S\setminus F\times F)\cup (F\times F)\). Also, the least nosp on \(S\) is characterized as the relation \(\rho^{-1}\), where \(\rho= \bigcup \sigma^n\) and \(a\sigma b\Leftrightarrow b^2\leq xay\) for some \(x,y\in S^1\).