# Decomposition of commutative ordered semigroups into archimedean components

## (*Lobachevskii Journal of Mathematics, Volume XXII*)

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.