Closed function sets on groups of prime order

TL;DR

This paper characterizes all function sets on prime order groups that are closed under certain compositions, revealing infinitely many non-finitely generated clones above a specific clone for primes greater than 2.

Contribution

It provides a complete classification of closed function sets and iterative algebras on prime order groups, and shows the existence of infinitely many non-finitely generated clones.

Findings

Complete description of closed function sets on prime order groups.

Identification of infinitely many non-finitely generated clones.

Implications for the structure of clones above a given clone.

Abstract

We give a full description of all sets of functions on the group $(\mathbb{ Z}_p, +)$ of prime order which are closed under the composition with the clone generated by $+$ from both sides. Thereby, we also get a description of all iterative algebras on $\mathbb{Z}_p$ which are closed under the composition with the clone generated by $+$ from both sides. As another consequence, there are infinitely many non finitely generated clones above $\operatorname{Clo}({\mathbb{ Z}_p \times \mathbb{ Z}_p, +})$ for $p > 2$.