Chains of maximum cardinality in distributive lattices

It's quite straightforward to construct a (complete) lattice in which no chain has maximum cardinality: for each $n\in \omega\setminus\{0\}$ let $C_n$ be a copy of $n$ with the chain ordering inherited from $n$. Put all the $C_n$'s side by side, and add a bottom and a top element.

However, this lattice is highly non-distributive, which leads to the following question:

Is there a distributive lattice $L$ such that for every chain $C\subseteq L$ there is a chain $C'\subseteq L$ such that $|C|<|C'|$?

• The union of an inclusion tower of chains is also a chain . As distributive lattices have a certain capacity for projecting an interval in to another, I don't see how you can avoid this tower. Gerhard "Will Try Chain Of Reasoning" Paseman, 2017.10.05. – Gerhard Paseman Oct 5 '17 at 17:05
• More importantly, are you assuming Zorn's Lemma? Gerhard "Or Perhaps Hausdorff's Maximum Principle?" Paseman, 2017.10.05. – Gerhard Paseman Oct 5 '17 at 17:12
• It's not about maximal chains (every chain is contained in a maximal one with respect to $\subseteq$, but about chains of maximum cardinality – Dominic van der Zypen Oct 5 '17 at 17:39
• Clearly the question is equivalent to the variant where you only consider maximal chains. In every modular lattice we have: if there exists a finite maximal chain, then all maximal chains have the same cardinality. Offhand I do not even know a modular lattice with two maximal chains of different cardinalities. – Goldstern Oct 7 '17 at 22:50
• @Goldstern: ${\mathcal P}(\omega)$. – Keith Kearnes Oct 8 '17 at 1:26

If the following conjecture of Don Monk is true, then it yields an affirmative answer to the question.

Conjecture. If $K$ is a nonempty set of infinite cardinals, then there is a Boolean algebra $A$ such that a cardinal $\kappa$ is the size of a maximal chain in $A$ if and only if $\kappa\in K$.

(If this conjecture is true, take $K=\{\aleph_n|\;n\in\omega\}$ and you have your example.)

This conjecture appears on the last page of

Monk, J. Donald, Towers and maximal chains in Boolean algebras. Algebra Universalis 56 (2007), no. 3-4, 337-347; doi:10.1007/s00012-007-2002-8, author's website.

In this paper, Monk proves a strong partial result:

Theorem. (GCH) If $K$ is a nonempty set of infinite regular cardinals, then there is a Boolean algebra $A$ with the property that a regular cardinal $\kappa$ is the size of a maximal chain in $A$ if and only if $\kappa\in K$.

Thus, Monk's Conjecture is that GCH is not needed in this theorem, and the restriction to regular cardinals is not needed.

• Thank you very much, in particular for the link to Monk's conjecture! – Dominic van der Zypen Oct 8 '17 at 9:04