2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210423We 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 $α$ of size $κ$, there is a set $\B$ of at most $2^κ$ join-semilattices, each one having a least element such that an algebraic lattice $L$ contains no chain of order type $I(α)$ 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 $[ω]^{<ω}$ belonging to $\B$, one is embeddable in all the others. We conjecture that if $α$ is countable, there is a finite set $\B$.11 pages, 2 figures, Proceedings ISOR'08, Algiers, Nov. 2-6, 2008CombinatoricsLogic06A12, 06B35text