  We study how the existence in an algebraic lattice $L$ of a chain of a given
type is reflected in the join-semilattice $K(L)$ of its compact elements. We
show that for every chain $\alpha$ of size $\kappa$, there is a set $\B$ of at
most $2^{\kappa}$ join-semilattices, each one having a least element such that
an algebraic lattice $L$ contains no chain of order type $I(\alpha)$ if and
only if the join-semilattice $K(L)$ of its compact elements contains no
join-subsemilattice isomorphic to a member of $\B$. We show that among the
join-subsemilattices of $[\omega]^{<\omega}$ belonging to $\B$, one is
embeddable in all the others. We conjecture that if $\alpha$ is countable,
there is a finite set $\B$.
