On chain decompositions of ordered semigroups (Q1775324)

A partially ordered semigroup \((S,\cdot,\leq)\) is called a natural ordered semilattice (chain) \(Y\) of ordered subsemigroups \(S_\alpha\) \((\alpha\in Y)\) if there is a semilattice (chain) congruence on \(S\), whose partition is \(\{S_\alpha\mid\alpha\in Y\}\) satisfying: \(a\in S_\alpha\), \(b\in S_\beta\), \(a\leq b\) (in \(S\)) \(\Rightarrow \alpha\leq\beta\) in \(Y\). By N. Kehayopulu and M. Tsingelis (1992), the least natural ordered semilattice congruence on a partially ordered semigroup \((S,\cdot,\leq)\) is given by: \(a\,{\mathcal N}\,b\) iff \(N(a)= N(b)\), where \(N(x)\) denotes the least filter of \(S\) containing \(x\in S\). Imposing different conditions on the components \(S_\alpha\) \((\alpha\in Y)\), numerous characterizations (at least 10 for each case) of natural ordered chains of ordered subsemigroups \(S_\alpha\) of the following kind are given: (i) \({\mathcal N}\)-simple, (ii) simple, (iii) \(J_n\)-simple, (iv) Archimedean. (Note that the concepts of ideal, filter resp. archimedean are understood in the semigroup- and order-theoretical sense).