On a stronger reconstruction notion for 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.Comment: 29 pp; Changes v1-->v2::typos corr.|L3.5+pf extended|Rem3.7 added|C. Pech found out that arg of L5.3-v1 solved Probl2-v1|L5.3, C5.4, Probl2 of v1 removed|C5.2, R5.4 new, contain parts of pf of L5.3-v1|L5.2-v1 is now L5.3,merged with concl of C5.4-v1,L5.3-v2 extends C5.4-v1|abstract, intro updated|ref[24] added|part of L5.3-v1 is L2.1(e)-v2, another part merged with pf of L5.2-v1 => L5.3-v