A single binary function is enough (Q2837212)

Let \(X\) be a non-empty set. Clones of finitary operations on \(X\) are considered. Let \({\mathcal O}\) denote the set of all finitary operations on \(X\). For any subset \(F\) of \({\mathcal O}\) the local clone generated by \(F\) consists of all operations on \(X\) that can be interpolated by operations from the clone generated by \(F\) on every finite set. The following results are proved: If \(X\) is infinite then every countable subset of \({\mathcal O}\) is included in a clone generated by a suitable binary operation. If \(X\) is countable then there exists a binary operation \(f\) on \(X\) such that the local clone generated by \(\{f\}\) equals \({\mathcal O}\). The local clone generated by a countable clone contains only countably many constant operations.NEWLINENEWLINEFor the entire collection see [Zbl 1234.08003].