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.