Rigid relational systems: A general approach (Q1922115)

Let \(A\) be a non-empty set, let \(H\) and \(F\) be clones of (finitary) operations on \(A\), let \(R\) denote the set of all (finitary) relations on \(A\), let \(Q\subseteq R\) and let \(\text{Pol }Q\) denote the set of all (finitary) operations on \(A\) which preserve all elements of \(Q\). \(Q\) is called \(H\)-polyrigid if \(\text{Pol } Q\subseteq H\). \(F\) is called \(H\)-minimal if it is not contained in \(H\), but every proper subclone of \(F\) is contained in \(H\). The main result of the paper says that if \(H\) is of the form \(\text{Pol } T\) for some finite subset \(T\) of \(R\) then \(Q\) is \(H\)-polyrigid iff \(\text{Pol }Q\) doesn't contain an \(H\)-minimal clone of operations on \(A\). The \(H\)-minimal clones are determined for some few special cases on a two- or three-element base set.