S\}$ ($S \in \{X,Y,Z\}$) and $\{u,v,w\}$ are fresh and mutually disjoint;

2.  $P$ has the following models:

    -   $U$;

    -   for each $I\subseteq X$, $J\subseteq Y$, $$M[I,J]=I\cup \overline{(X\setminus I)}\cup J\cup \overline{(Y\setminus J)}\cup Z\cup \overline{Z}\cup \{u,v\}$$ and $$M'[I,J]=I\cup \overline{(X\setminus I)}\cup J\cup \overline{(Y\setminus J)}\cup Z\cup \overline{Z}\cup \{v,w\};$$

3.  for each $I\subseteq X$, $J\subseteq Y$, the models[^2] of the reduct $P^{M[I,J]}$ are $M[I,J]$ and $$O[I] = I\cup \overline{(X\setminus I)};$$

4.  for each $I\subseteq X$, $J\subseteq Y$, the models of the reduct $P^{M'[I,J]}$ are $M'[I,J]$ and

    -   for each $K\subseteq Z$ such that $I\cup J \cup K\not \models \phi$, $$N[I,J,K] = I \cup \overline{(X\setminus I)}\cup J\cup \overline{(Y\setminus J)}\cup K\cup \overline{(Z\setminus K)}\cup \{v\};$$

5.  the models of the reduct $P^U$ are $U$ itself, $M[I,J]$, $M'[I,J]$, and $O[I]$, for each $I\subseteq X$, $J\subseteq Y$, and $N[I,J,K]$ for each $I\subseteq X$, $J\subseteq Y$, $K\subseteq Z$, such that $I\cup J \cup K\not \models \phi$.

=\[\] =\[draw,thick,-&gt;\] =\[draw,thick\] =\[draw,thick,rectangle, rounded corners\]

\(U) at (0,1) [$U$]{}; (MI0J0) at (-4,-1) [$M[I_0,J_0]$]{}; (dots1) at (-2.5,-1) [$\cdots$]{}; (MImJn) at (-1,-1) [$M[I_m,J_n]$]{}; (MpI0J0) at (1,-1) [$M'[I_0,J_0]$]{}; (dots2) at (2.5,-1) [$\cdots$]{}; (MpImJn) at (4,-1) [$M'[I_m,J_n]$]{};

\(U) edge node\[above left\] [$\subset$]{} (MI0J0); (dots3) at (-1.5,-0.3) [$\cdots$]{}; (U) edge node\[left\] [$\subset$]{} (MImJn);

\(U) edge node\[right\] [$\subset$]{} (MpI0J0); 