A characterization of well-founced algebraic lattices

We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice L is well-founded if and only if K(L), the join-semilattice of compact elements of L, is well-founded and contains neither [ω], nor Ω(ω∗) as a join-subsemilattice. As an immediate corollary, we get that an algebraic modular lattice L is well-founded if and only if K(L) is wellfounded and contains no infinite independent set. If K(L) is a joinsubsemilattice of I<ω(Q), the set of finitely generated initial segments of a well-founded poset Q, then L is well-founded if and only if K(L) is well-quasi-ordered.