Minimal operations over permutation groups

TL;DR

This paper classifies minimal operations over permutation groups, reducing the possible types above non-trivial groups and resolving open questions in the field.

Contribution

It refines the classification of minimal operations over permutation groups and addresses previously open questions, especially for oligomorphic groups.

Findings

At most four types of minimal operations above any non-trivial permutation group.

Except for Boolean groups acting freely, only three types exist.

One of Bodirsky and Chen's classified types does not occur for oligomorphic groups.

Abstract

We classify the possible types of minimal operations above an arbitrary permutation group. Above the trivial group, a theorem of Rosenberg yields that there are five types of minimal operations. We show that above any non-trivial permutation group there are at most four such types. Indeed, except above Boolean groups acting freely on a set, there are only three. In particular, this is the case for oligomorphic permutation groups, for which we improve a result of Bodirsky and Chen by showing one of the types in their classification does not exist. Building on these results, we answer three questions of Bodirsky that were previously open.