On sets related to maximal clones. (Q2861068)

The concept of an operation \(f\colon A^n\to A\) preserving a given relation \(\rho\in A^k\) is an important tool in the theory of clones. The authors modify this concept by including some parameters from the base set \(A\). The resulting sets of operations are not clones, but they enable to express the clone of all \(\rho\)-preserving operations by means of union and intersection. For instance, if \((A,\leq)\) is a finite partially ordered set, then the set of all \(n\)-ary order-preserving functions can be expressed using the sets \(P_{\overline a,b}^{n,1}\) and \(P_{\overline a,b}^{n,2}\), with \(\overline a\in A^n\), \(b\in A\). An \(n\)-ary operation \(f\) belongs to \(P_{\overline a,b}^{n,1}\) if \(\overline a\leq\overline x\) implies \(b\leq f(\overline x)\). (The other set is defined dually.)NEWLINENEWLINE Similar results are proved for operations preserving equivalence relations and central relations.