On a stronger reconstruction notion for monoids and clones

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