Non-distributive lattices with chains of different lengths

Question

Consider the following Hasse diagramme. We will dub such lattices $\mathbf{Mkn}$.

Assume a propositional language over $\{\wedge,\vee,\neg\}$. Let $v$ be a mapping from the set of all propositional variables to $\{\top,\ldots,\bot\}$, $\wedge$ and $\vee$ coincide with the meet and join while $\neg$ works as follows ($\phi$ and $\chi$ are formulas over $\{\wedge,\vee,\neg\}$).

1. $\forall v:v(\neg(\phi\wedge\chi))=v(\neg\phi\vee\neg\chi)\quad\forall v:v(\neg(\phi\vee\chi))=v(\neg\phi\wedge\neg\chi)$.
2. $\forall v:v(\neg\neg\phi)=v(\phi)$.
3. If $v(\phi\vee\neg\phi)=\top$, then $v(\phi)\in\{\top,\bot\}$.

Is it true that the following statement holds?

For any fixed $\mathbf{n}>2$ if there a valuation $v$ on $\mathbf{Mkn}$ such that $v(\phi)\not\leqslant v(\chi)$, then there is a valuation $v'$ on $\mathbf{M1n}$ such that $v'(\phi)\not\leqslant v'(\chi)$.

It seems to me that it is true. The argument goes as follows.

First, it is easy to show (cf. Belnap's Four-Valued Logic and De Morgan Lattices by J.M. Font for a method) that if we take $D$ as an ultrafilter on any $\mathbf{Mkn}$ lattice, then $$\forall v:v(\phi)\in D\Rightarrow v(\chi)\in D\Longleftrightarrow\forall v:v(\phi)\leqslant v(\chi)$$

Now take a valuation $v$ on $\mathbf{Mkn}$ such that $v(\phi)\in D$ but $v(\chi)\notin D$.

Construct $v'$ as follows.

1. If $v(p)=\top$ or $v(p)=\bot$, then $v'(p)=\top$ or $v'(p)=\bot$, respectively.
2. Otherwise, $v'(p)\in\{\mathbf{1},\ldots,\mathbf{n}\}$ and the following conditions apply: if $v(p\vee q)=\top$, then $v'(p)\neq v'(q)$; if $v(p\vee q)\neq\top$, then $v'(p)=v'(q)$; if $v(p)\in D_{\mathbf{Mkn}}$, then $v'(p)\in D_{\mathbf{M1n}}$.

One can show by induction that $v(\phi)\in D_{\mathbf{Mkn}}$ iff $v'(\phi)\in D_{\mathbf{M1n}}$.

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However, I am not sure and strongly suspect a mistake in the proof.

Answer 1

I've tried translating your problem into a more universal algebraic question. Hopefully I've translated correctly.

Let $\mathbf{M}_{k,n}$ be the lattice obtained by taking $n$ disjoint chains of length $k$ and adjoining a minimum element and a maximum element.

Question: If $\mathbf{M}_{1,n}$ satisfies the identity $\phi\approx\chi$, must $\mathbf{M}_{k,n}$ satisfy the identity $\phi\approx\chi$?

The answer is no. The lattice $\mathbf{M}_{1,n}$ is modular for every $n$, but $\mathbf{M}_{k,n}$ is not modular whenever $k>1$ and $n>1$.