The length of chains in algebraic lattices

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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 α of size κ, there is a set of at most 2κ 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. We show that among the join-subsemilattices of [ω]<ω belonging to, one is embeddable in all the others. We conjecture that if α is countable, there is a finite set.