For $\kappa>0$ a regular cardinal, let $Ht_\kappa$ denote the following partial quasi-order:
 
(i) elements(objects) of $Ht_\kappa$ are classes of sets of size $\kappa$
with the property that,
 ($<_\kappa$) for every $S\subseteq X$ of $|S| <\kappa$, there is $x_S\in X$ and $a_S, |a_S|<\kappa$,
such that $\cup S\subseteq x_S\cup  a_S$.

$X\leq_\kappa Y$ in $Ht_\kappa$ iff for every $x\in X$ exists $y\in Y$  such that $|x\setminus y|<\kappa$

To avoid set-theoretic difficulties, you may want instead to consider $Ht_\kappa(\lambda)$
---the suborder consisting only of classes of subsets of $\lambda$---for some big cardinal $\lambda$.

It is easy to prove that $\kappa$ is measurable or countable iff $Ht_\kappa$ is not dense. (Proof : $I\leq_\kappa \{\kappa\}$
means that $I$ is a $\kappa$-closed ideal on $\kappa$; $I$ is maximal such iff there is nothing strictly $<_ \kappa$-between
$I$ and $\{\kappa\}$. Finally, note that there is nothing between $X\leq_\kappa Y$ implies that for every $y\in Y$, there is
nothing in between {$x\cap y:x\in X$ }$\leq_\kappa ${ $y$} and $X<_\kappa Y$ implies that at least one of these inequalities is strict).

Hence one can ask:

>Is it consistent that $Ht_\omega$ is elementary equivalent to $Ht_\kappa$ for some $\kappa>\omega$ (in the language of
partial orders)? What is this on the scale of large cardinals?


> Does there exist a "canonical" monotone function $\mu:Ht_\kappa \longrightarrow \{0,1\}$
into the two-element order  0<1 ?

"Canonical" here means that for every automorphism $s: Ht_\kappa\longrightarrow Ht_\kappa$, $\mu(s(X))=\mu(X)$;
"continious" means that for every set $X$, $\mu(\sup_i X_i)=\sup_i\mu(X_i)$ and $\mu(\inf_i X_i)=\inf_i\mu(X_i)$ (whenever these are well-defined).

If the latter is too strong, one may require the same only for directed $X$. (Note that
$sup X$  and $inf X$  exist in $Ht_\kappa$ for every $X$). I can think of
only functions which are cut-offs of  $\Bbb Lcard (X)=\min$ { $|Y|: Y\geq X$}...