Bodirsky, M., Evans, D., Kopatscher, M., & Pinsker, M. (2018). A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoids. Israel Journal of Mathematics, 224(1), 57–82. https://doi.org/10.1007/s11856-018-1645-9
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 language which can neither be reconstructed up to first-order biinterpretations 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.
