Decomposition of commutative ordered semigroups into archimedean components

The decomposition of a commutative semigroup (without order) into its archimedean components, by means of the division relation, has been studied by Clifford and Preston. Exactly as in semigroups, the complete semilattice congruence "$\mathcal N$" defined on ordered semigroups by means of filters, plays an important role in the structure of ordered semigroups. In the present paper we introduce the relation "$\eta$" by means of the division relation (defined in an appropriate way for ordered case), and we prove that, for commutative ordered semigroups, we have $\eta=\mathcal N$. As a consequence, for commutative ordered semigroups, one can also use that relation $\eta$ which has been also proved to be useful for studying the structure of such semigroups. We first prove that in commutative ordered semigroups, the relation $\eta$ is a complete semilattice congruence on $S$. Then, since $\mathcal N$ is the least complete semilattice congruence on $S$, we have $\eta=\mathcal N$. Using the relation $\eta$, we prove that the commutative ordered semigroups are, uniquely, complete semilattices of archimedean semigroups which means that they are decomposable, in a unique way, into their archimedean components.