Complete chains of semigroups play an essential
role in the decomposition of ordered semigroups. In this paper we
prove that an ordered semigroup $S$ is a complete chain of
semigroups of a given type, say $\cal T$, if and only if it is
decomposable into pairwise disjoint subsemigroups $S_\alpha$ of
$S$ indexed by a semilattice $Y$ satisfying, for any $\alpha,
\beta\in Y$, the two conditions $S_\alpha S_\beta\subseteq
S_{\alpha \beta}$ and whenever $S_\alpha \cap
(S_\beta]\not=\emptyset$, then $\alpha=\alpha \beta
(=\beta\alpha)$.