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 " 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 "η" by means of the division relation (defined in an appropriate way for ordered case), and we prove that, for commutative ordered semigroups, we have η = N. As a consequence, for commutative ordered semigroups, one can also use that relation η 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 η is a complete semilattice congruence on S. Then, since N is the least complete semilattice congruence on S, we have η = N. Using the relation η, 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.
|DVI format||PostScript format||PDF format||MathML Format|