Stability of the Souslin operation (Q800390)

If I, J, K are sets, \(\tau\) :\(I\times J\to K\) is a function and L is a complete lattice, then \(S:L^ K\to L\), defined by \(S(A)=\vee_{j\in J}\wedge_{i\in I}A(\tau (i,j)), A\in L^ K\), is the \(\tau\) -Souslin operation. For \(E\subset L\), \({\mathcal S}(E)\) is defined by \({\mathcal S}(E)=\{x\in L| \exists A[A:K\to E,\quad x=S(A)]\}.\) The following theorem is proved. For \(\tau\) a Souslin substitution and L a complete, completely distributive lattice, \({\mathcal S}:{\mathcal P}(L)\to {\mathcal P}(L)\) is a closure operator.