Chain decomposition of ordered semigroups (Q1611019)

A partially ordered (p.o.) semigroup \((S,\cdot ,\leq)\) is called a natural ordered semilattice, in particular, chain \(Y\) of subsemigroups \(S_\alpha \), \((\alpha\in Y)\) if \(a\leq b\), \(a\in S_\alpha\), \(b\in S_\beta\) imply that \(\alpha\leq \beta\) in \(Y\). If \(\rho\) is any binary relation on \(S\) then \(S\) is said to be \(\rho\)-simple if \(\rho\) is the universal relation. Using several particular relations \(\rho\) defined earlier by the author, conditions for \((S, \cdot ,\leq)\) are given to be a natural ordered chain of \(\rho\)-simple subsemigroups. The definitions of the \(\rho\)'s considered involve left, right, two-sided, or bi-ideals, where ``ideal'' is understood as semigroup and order-ideal. In particular, chain decompositions into \(t\)-simple subsemigroups \(S_\alpha\) are studied (as a generalization of semilattices of groups in the non-ordered case): \((S_\alpha,\cdot, \leq)\) is \(t\)-simple if \(S_\alpha\) is \({\mathcal L}\)- and \({\mathcal R}\)-simple, where \(a{\mathcal L}b\) if and only if the ``left-ideals'' of \(S_\alpha\) generated by \(a\) and \(b\), respectively, are equal \(({\mathcal R}\) is defined dually). Also, the case of 1-\((t\)-)archimedian components is considered: \((S_\alpha, \cdot,\leq)\) is 1-\((t\)-)archimedian if for all \(a,b\in S_\alpha\), \(b^m\leq xa\) \((b^m\leq axa)\) for some \(m\in\mathbb{N}\), \(x\in S\). Up to 15 equivalent conditions for a p.o. semigroup to be a natural ordered chain of subsemigroups of one of the above \(\rho\)-simple kinds are given.