I am looking for a Kripke frame condition corresponding to the McKinsey axiom M: $\Box\Diamond p \rightarrow \Diamond\Box p$. I read somewhere the following condition:
"For every partitioning of the set of worlds into two disjoint partitions, every world can see a world whose successors lie all in the same partition."
This follows from rewriting the formula as $\Diamond\Box \lnot p \lor \Diamond\Box p$. But it is difficult to use because it involves sets of worlds.
So I am looking for a frame condition for KM of the form $\forall w P(w)$, where $P$ is a predicate expressing visibility between $w$ and other worlds. In the case of M, this condition cannot be first-order, but it could still be second order. For example:
Formula $(p\land \Box(\Diamond p \rightarrow p)) \rightarrow \Box p$ corresponds to frames for which $\forall w$, if $wRw'$ then there is a finite sequence $w_0,...,w_n$ such that $w_0=w'$, $w_0Rw_1Rw_2...Rw_nRw$ and also $wRw_i$ for $1\le i \le n$. See article A Simple Incomplete Extension of T which is the Union of Two Complete Modal Logics with f.m.p. by Roy A. Benton.
Formula $\Box(\Box p \rightarrow p) \rightarrow \Box p$ (Loeb) corresponds to frames for which $\forall w$ we have $wRw' \land w'Rw'' \rightarrow wRw''$ (transitive) and also there is no infinite sequence of worlds $wRw_1Rw_2R...$ starting from $w$ (converse well-founded). See P. Blackburn Modal Logic pp 131; it is also shown there that both the Loeb and the McKinsey formulas do not correspond to a first order condition.
The above examples are not first-order conditions. But note that they describe their class of frames by stating what an arbitrary world can see, i.e. without using a partition or a valuation.
So my question is: is there a similar frame condition known for axiom M?
This should correspond to the frames of KM itself, i.e not in conjunction with other axioms. My hope is that in such a form it would be better suited for analyzing the extensions of KM − any extension, not just K4M.