(dots4) at (1.5,-0.3) [$\cdots$]{}; (U) edge node\[above right\] [$\subset$]{} (MpImJn);

(CMMI0J0) at (-4,-3) [$O[I_0]$]{}; (MI0J0) edge node\[left\] [$P^{M[I_0,J_0]}$]{} (CMMI0J0);

(dots5) at (-2.75,-3) [$\cdots$]{};

(CMMImJn) at (-1.5,-3) [$O[I_m]$]{}; (MImJn) edge node\[left\] [$P^{M[I_m,J_n]}$]{} (CMMImJn);

(CMMpI0J0) at (1,-3) [$
  \begin{array}{l}
   N[I_0,J_0,K] \mbox{\ s.t.}\\
   I_0 \cup J_0 \cup K \not \models \phi  
  \end{array}
$]{}; (MpI0J0) edge node\[right\] [$P^{M'[I_0,J_0]}$]{} (CMMpI0J0);

(dots5) at (3,-3) [$\cdots$]{};

(CMMpImJn) at (5,-3) [$
  \begin{array}{l}
   N[I_m,J_n,K] \mbox{\ s.t.}\\
   I_m \cup J_n \cup K \not \models \phi  
  \end{array}
 $]{}; (MpImJn) edge node\[right\] [$P^{M'[I_m,J_n]}$]{} (CMMpImJn);

(CMU) at (0,3) [$
  \begin{array}{c}
   M[I_0,J_0]\ \cdots\ M[I_m,J_n]\qquad M'[I_0,J_0]\ \cdots\ M'[I_m,J_n]\\[2ex]  
   O[I_0]\ \cdots\ O[I_m] \qquad
   \begin{array}{l}
   N[I_0,J_0,K] \mbox{\ s.t.}\\
   I_0 \cup J_0 \cup K \not \models \phi  
  \end{array}
\ \cdots\ 
  \begin{array}{l}
   N[I_m,J_n,K] \mbox{\ s.t.}\\
   I_m \cup J_n \cup K \not \models \phi  
  \end{array}
  \end{array}
$]{};

\(U) edge node\[right\] [$P^{U}$]{} (CMU);

The structure of models of $\Phi$-reductions and the “countermodels” (see below what we mean by this term) of the relevant reducts is sketched in Figure \[fig:phired\]. The center of the diagram contains the models of the $\Phi$-reduction and their subset relationship. For each of the model the respective box lists the “countermodels,” by which we mean those reduct models which can serve as counterexamples for the original model 