The length of chains in algebraic lattices
Ilham Chakir, Maurice Pouzet · arXiv (Cornell University) · 2008
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 $\B$ 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]^{