On the local closure of clones on countable sets

Abstract

We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function $f:\mathbb{N}\to \mathbb{N}$ such that the $n$-ary part of $C$ is equal to the $n$-ary part of $\mathrm{Pol}{\mathrm{Inv}}^{[f(n)]}C$, where ${\mathrm{Inv}}^{[f(n)]}C$ denotes the set of $f(n)$-ary invariant relations of $C$.