families of mutually independent unpushed pure $\sigma$-buttons and $\sigma$-switches over $M$. Then
(1) any inverted lollipop can be $\Gamma$-labelled over $M$, and
(2) $ML\Gamma^M$ is contained within S4.2Top.
PROOF. We show (1). Claim (2) then follows from the conjunction of Theorems 6 and 12.
Let $L$ be a frame which is an inverted lollipop. Let $F$ be the quotient partial order of $L$ under the natural equivalence relation. Then $F$ is a finite Boolean algebra with a single extra node on top; in particular, $F$ is a topped partial order. Therefore, the partial order of non-maximal elements of $F$ is isomorphic to the power set algebra $\wp(A)$ for some finite set $A$. We fix such a set $A$.
With each element $a \in F$ which is not maximal, there is associated a cluster $w_1^a, w_2^a, \dots, w_{k_a}^a$ of worlds of $L$. By adding dummy nodes to each cluster, we may assume that there is some natural number $m$ such that for each non-maximal $a \in F$, the sizes $k_a$ of the complete clusters at node $a$ are the same, and equal to $2^m$. Also, suppose that $F$ has size $2^n+1$, that is, the size of $A$ is $n$, so there are $n$ atoms in the Boolean algebra of non-maximal elements of $F$. We can therefore think of the Boolean algebra of non-maximal elements of $F$ as the worlds $w_j^a$ where $a \subseteq A$, and $j < 2^m$, with the order obtained by $w_j^a \le w_i^c$ if and only if $a \subseteq c$. Also, since $F$ consists exactly of this pre-Boolean algebra and a single extra node above every element of it, we can consider $F$ as being made up of worlds $w_j^a$ where $a \subseteq A$, and $j < 2^m$, with the order obtained by $w_j^a \le w_i^c$ if and only if $a \subseteq c$, and a node $t$ with $w_j^a < t$ for each $a$ and $j$.
Associate with each element $i \in A$ an unpushed pure $\sigma$-button $b_i$ such that the collection ${b_i | i \in A}$ form a mutually independent family with $m$-many $\sigma$-switches $s_0, s_1, \dots, s_{m-1}$. Without loss of generality, we can assume that all the $\sigma$-switches are off in $M$. For $j < 2^m$, let $\bar{s}_j$ be the assertion that the pattern of switches corresponds to the binary digits of $j$ (i.e., $s_k$ is true if and only if the $k$th binary bit of $j$ is 1). We associate the node $w_j^a$ with the assertion
and we associate the node $t$ with the assertion $\Phi_t = \sigma$. Now, if $W$ is a model in the multiverse of $\Gamma$ generated by $M$, and $W \models \Phi_{w_j^a}$, then by the mutual independence of buttons and switches combined with our remark that pushing these buttons cannot push the button $\sigma$, we see that $W \models \Diamond \Phi_{w_r^c}$ if and only if $a \subseteq c$.
Also, for any model $W$ in the multiverse of $M$ generated by $\Gamma$, if $W \models \neg \sigma$, then as $\sigma$ is itself a button, it follows that $W \models \Diamond \Phi_t$. Therefore, if $W \models \Phi_{w_j^a}$, then $W \models \Diamond \Phi_t$. Also, since all of these buttons and switches are off in $M$, we have $M \models \Phi_{w_0^\varphi}$. Thus, we have provided a $\Gamma$-labelling of this frame for $W$, hence demonstrating that we can label all inverted lollipops. The result follows. ∨