Abstract
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.
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The first author has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 257039).
The third author has been funded through project P27600 of the Austrian Science Fund (FWF).
The fourth author has been funded through projects I836-N23 and P27600 of the Austrian Science Fund (FWF).
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Bodirsky, M., Evans, D., Kompatscher, M. et al. A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoid. Isr. J. Math. 224, 57–82 (2018). https://doi.org/10.1007/s11856-018-1645-9
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DOI: https://doi.org/10.1007/s11856-018-1645-9