A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoid

Manuel Bodirsky, David Evans, Michael Kompatscher, Michael Pinsker
2018 Israel Journal of Mathematics  
We present an example of two countable ω-categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids -in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable ω-categorical structure in a finite relational
more » ... ge which can neither be reconstructed up to first-order bi-interpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone. M. BODIRSKY, D. EVANS, M. KOMPATSCHER, AND M. PINSKER are stable under bi-interpretability. Hence, we focus on a subproblem: is it true that when Aut(A) and Aut(B) are isomorphic as groups, then they are also isomorphic as topological groups? Rather surprisingly, isomorphisms between automorphism groups of countable structures are typically homeomorphisms. And in fact, it is consistent with ZF + DC that all homomorphisms between closed subgroups of Sym(ω) are continuous, and that all isomorphisms between closed subgroups of Sym(ω) are homeomorphisms; see the end of Section 3.2 for more explanation. Using the existence of non-principal ultrafilters on ω, it is relatively easy to show that there are oligomorphic permutation groups with non-continuous homomorphisms to Z 2 . But it was open for a while whether for countable ω-categorical structures A and B the existence of an isomorphism between Aut(A) and Aut(B) implies the existence of an isomorphism which is additionally a homeomorphism. This problem was solved by the second author and Hewitt [EH90], by giving two structures A and B for which the answer was negative. Natural objects that carry more information about a structure A than Aut(A) are its endomorphism monoid End(A), which consists of the set of homomorphisms from A to A, or, even more generally, its polymorphism clone Pol(A), which consists of the set of all homomorphisms from A k to A, for all k ≥ 1. We are going to show the following theorem related to results of Lascar [Las89]; see also the discussion in Section 4. Theorem 1.1. There are countable ω-categorical structures A, B such that End(A) and End(B) are isomorphic, but not topologically isomorphic.
doi:10.1007/s11856-018-1645-9 fatcat:i2v7fa3ei5fazm4rb2swkqjwh4