# On a stronger reconstruction notion for monoids and clones

## TL;DR

This paper introduces a new reconstruction concept called automatic action compatibility for monoids and clones, providing a framework to analyze their structure and symmetry properties, with applications to various well-known countable structures.

## Contribution

It defines automatic action compatibility, characterizes automatic homeomorphicity for transformation monoids, and extends these concepts from groups to monoids and clones under weak conditions.

## Key findings

- Established automatic action compatibility for several countable structures.
- Provided a characterization of automatic homeomorphicity for transformation monoids.
- Extended the framework from groups to monoids and clones.

## Abstract

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.