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arXiv · 0812.2300

A characterization of well-founded algebraic lattices

Ilham Chakir·Maurice Pouzet
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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 $[ω]^{<ω}$, nor $\underlineΩ(ω^*)$ 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 well-founded and contains no infinite independent set. If $K(L)$ is a join-subsemilattice 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.

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Ilham Chakir, Maurice Pouzet. 2008-12-12. A characterization of well-founded algebraic lattices. https://arxiv.org/abs/0812.2300

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