## The length of chains in algebraic lattices

Chakir, Ilham
Pouzet, Maurice
##### Description
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 $\B$ 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 $\B$. We show that among the join-subsemilattices of $[\omega]^{<\omega}$ belonging to $\B$, one is embeddable in all the others. We conjecture that if $\alpha$ is countable, there is a finite set $\B$.
Comment: 11 pages, 2 figures, Proceedings ISOR'08, Algiers, Nov. 2-6, 2008
##### Keywords
Mathematics - Combinatorics, Mathematics - Logic, 06A12, 06B35