On the local closure of clones on countable sets (Q1686327)

The author studies clones on infinite sets. For a clone \(C\) on a set \(A\), its local closure consists of all those finitary operations that can be interpolated at each finite subset of their domain by a function in \(C\) and the local closure of \(C\) is equal to \(\operatorname{Pol} \operatorname{Inv} C\). A clone is called locally closed if it is equal to its local closure. Theorem 1.1 states that a clone with quasigroup operations on a countable infinite set is either locally closed, or its local closure \(\operatorname{Pol} \operatorname{Inv} C\) is uncountable. Theorem 1.2 states that there exists an infinite countable set \(A\) and a constantive clone \(C\) on \(A\) such that \(\operatorname{Pol} \operatorname{Inv} C\) differs from \(C\) but the cardinality of \(\operatorname{Pol} \operatorname{Inv} C\) is countable. Theorem 1.3 gives necessary and sufficient conditions under which \(\operatorname{Pol} \operatorname{Inv} C\) is at most countable for a clone \(C\) on a countable infinite set \(A\). A slightly modified theorem is derived for a constantive clone \(C\) with quasigroup operations.