[Paper Review] The length of chains in algebraic lattices

Ilham Chakir, Maurice Pouzet|ArXiv.org|Dec 11, 2008

Advanced Algebra and Logic9 references3 citations

TL;DR

This paper investigates the relationship between the length of chains in algebraic lattices and the structure of their join-semilattices of compact elements. It proves that for any order type α, the absence of chains of type I(α) in an algebraic lattice L is characterized by the absence of certain join-subsemilattices in K(L), with a bound of at most 2^|α| such forbidden subsemilattices. For countable α, the authors conjecture a finite characterization exists.

ABSTRACT

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]^{

Motivation & Objective

• To determine how the presence or absence of chains of a given order type α in an algebraic lattice L is reflected in the join-semilattice K(L) of its compact elements.
• To generalize results from ideal lattices of posets to algebraic lattices via join-semilattices with least elements.
• To investigate whether, for countable order types α, there exists a finite set of forbidden join-subsemilattices that characterize the absence of chains of type I(α) in K(L).
• To explore structural properties of join-semilattices such as Ω(α′) and their role in embedding chains into ideal lattices.

Proposed method

• The authors define the class 𝕁 of join-semilattices with a least element and use the notion of Forb(ℬ) to denote those in 𝕁 avoiding join-subsemilattices isomorphic to members of a set ℬ.
• They introduce the class 𝕁_¬α of join-semilattices P for which J(P) contains no chain of type I(α), and show that 𝕁_¬α = Forb(ℬ) for some ℬ ⊆ 𝕁 with |ℬ| ≤ 2^|α|.
• They use sierpinskizations of chains and monotonic embeddings to construct specific join-semilattices, such as Ω(α′), which are universal for certain chain types.
• They prove that for α = ωα′ + n with n < ω, the structure Q_α = I_<ω(S_α) with S_α = Ω(α′) ⊕ n is embeddable into I_<ω(S) for any sierpinskisation S of α and ω.
• They establish that Q_α ∈ 𝕁_α and that embeddings preserve finite joins, linking the structure of K(L) to the existence of long chains in L.
• They rely on known results from [11] on sierpinskizations and embeddings, particularly that Ω(α′) embeds into any sierpinskisation of ωα′ and ω.

Experimental results

Research questions

• RQ1 For a given order type α, what is the minimal set of join-subsemilattices whose absence in K(L) characterizes the absence of chains of type I(α) in the algebraic lattice L?
• RQ2 Can the bound of 2^|α| forbidden subsemilattices be improved to a finite set when α is countable?
• RQ3 How do sierpinskizations of chains relate to the embedding of join-semilattices into ideal lattices of posets?
• RQ4 What structural properties of Ω(α′) and related constructions ensure their universality in characterizing chain types?
• RQ5 Is there a canonical join-subsemilattice in the family of forbidden subsemilattices for a given α, embeddable in all others?

Key findings

• For every order type α, there exists a set ℬ of at most 2^|α| join-semilattices in 𝕁 such that an algebraic lattice L contains no chain of order type I(α) if and only if K(L) contains no join-subsemilattice isomorphic to any member of ℬ.
• Among the join-subsemilattices of [ω]^{<ω} in the forbidden family ℬ for α = ω*, one is embeddable in all others, indicating a canonical minimal obstruction.
• For α = ω or finite, the forbidden family ℬ has a single member, namely the chain α′ with α = 1 + α′, showing a simpler structure in these cases.
• For α = ω* or η, the forbidden family ℬ has exactly two members: α and Ω(α), confirming a known result from [11] in the new setting.
• The construction Q_α = I_<ω(S_α) with S_α = Ω(α′) ⊕ n is shown to be embeddable into I_<ω(S) for any sierpinskisation S of α and ω, and Q_α ∈ 𝕁_α, proving that such structures are sufficient to detect the presence of I(α) in J(P).
• The authors conjecture that if α is countable, then a finite set ℬ suffices, suggesting a potential strengthening of the 2^|α| bound.