A characterization of well-founded algebraic lattices

TL;DR

This paper characterizes well-founded algebraic lattices using forbidden subsemilattices of their compact elements, providing criteria for well-foundedness based on the structure of these compact elements.

Contribution

It introduces a new characterization of well-founded algebraic lattices through forbidden subsemilattices, linking lattice properties to the structure of compact elements.

Findings

An algebraic lattice is well-founded iff its compact elements form a well-founded join-semilattice without specific subsemilattices.

A modular algebraic lattice is well-founded iff its compact elements are well-founded and contain no infinite independent set.

If compact elements are finitely generated initial segments of a well-founded poset, then the lattice is well-founded iff the compact elements are well-quasi-ordered.

Abstract

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 $[\omega]^{<\omega}$, nor $\underline\Omega(\omega^*)$ 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 well-founded and contains no infinite independent set. If $K(L)$ is a join-subsemilattice of $I_{<\omega}(Q)$, 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.