The length of chains in algebraic lattices

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 $\text{B}$ 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 $\text{B}$. We show that among the join-subsemilattices of $[\omega {]}^{<\omega }$ belonging to $\text{B}$, one is embeddable in all the others. We conjecture that if $\alpha$ is countable, there is a finite set $\text{B}$.