On the local closure of clones on countable sets

TL;DR

This paper investigates the structure of clones on countable sets with quasigroup operations, showing that their local closure can be characterized by a function relating arity and invariant relations.

Contribution

It establishes a new characterization of locally closed clones with quasigroup operations on countable sets via a function linking arity and invariant relations.

Findings

Existence of a function f linking n-ary parts to invariant relations

Characterization of locally closed clones with quasigroup operations

Insight into the structure 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 $C$.