Abstract
Every transformation monoid comes equipped with a canonical topology, the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever \({\mathbf {A}}\) is a countable \(\aleph _0\)-categorical G-finite structure whose automorphism group has a trivial center and if \({\mathbf {B}}\) is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism.
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References
Barham, R., Automatic homeomorphicity of locally moving clones, ArXiv e-prints, 2015.
Behrisch, M., J. K. Truss, and E. Vargas-García, Reconstructing the topology on monoids and polymorphism clones of the rationals, Studia Logica 105(1): 65–91, 2017.
Bodirsky, M., D. Evans, M. Kompatscher, and M. Pinsker, A counterexample to the reconstruction of \(\omega \)-categorical structures from their endomorphism monoids, ArXiv e-prints, 2015.
Bodirsky, M., and M. Pinsker, Topological Birkhoff, Trans. Amer. Math. Soc. 367(4): 2527–2549, 2015.
Bodirsky, M., M. Pinsker, and A. Pongrácz, Reconstructing the topology of clones, Trans. Amer. Math. Soc. 369(5): 3707–3740, 2017.
Cameron, P. J., and J. Nešetřil, Homomorphism-homogeneous relational structures, Combin. Probab. Comput. 15(1-2): 91–103, 2006.
Dixon, J. D., P. M. Neumann, and S. Thomas, Subgroups of small index in infinite symmetric groups, Bull. London Math. Soc., 18(6): 580–586, 1986.
Dolinka, I., and D. Mašulović, Properties of the automorphism group and a probabilistic construction of a class of countable labeled structures, Journal of Combinatorial Theory, Series A 119(5): 1014–1030, 2012.
Evans, D. M., and P. R. Hewitt, Counterexamples to a conjecture on relative categoricity, Ann. Pure Appl. Logic 46(2): 201–209, 1990.
Herwig, B., Extending partial isomorphisms for the small index property of many \(\omega \)-categorical structures, Israel J. Math. 107: 93–123, 1998.
Hodges, W., Model theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993.
Hodges, W., I. Hodkinson, D. Lascar, and S. Shelah, The small index property for \(\omega \)-stable \(\omega \)-categorical structures and for the random graph, J. Lond. Math. Soc., II. Ser. 48(2): 204–218, 1993.
Kechris, A. S., and C. Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proceedings of the London Mathematical Society 94(2): 302–350, 2007.
Lascar, D., Le demi-groupe des endomorphismes d’une structure \(\aleph _0\)-catégorique, in: M. Giraudet, (ed.). Actes de la Journée Algèbre Ordonnée (Le Mans, 1987) 1989, pp. 33–43.
Lascar, D., Autour de la propriété du petit indice. (On the small index property)., Proc. Lond. Math. Soc., III. Ser. 62(1): 25–53, 1991.
Macpherson, D., A survey of homogeneous structures, Discrete Math. 311(15): 1599–1634, 2011.
Pech, C., and M. Pech, Polymorphism clones of homogeneous structures (universal homogeneous polymorphisms and automatic homeomorphicity), ArXiv e-prints, 2015.
Pech, C., and M. Pech, On automatic homeomorphicity for transformation monoids, Monatsh. Math. 179(1): 129–148, 2016.
Pöschel, R., A general Galois theory for operations and relations and concrete characterization of related algebraic structures, Report (Akademie der Wissenschaften der DDR. Zentralinstitut für Mathematik und Mechanik), Akademie der Wissenschaften der DDR, 1980.
Rosendal, C., Automatic continuity of group homomorphisms, Bull. Symbolic Logic 15(2): 184–214, 2009.
Solecki, S., Extending partial isometries, Israel J. Math. 150: 315–331, 2005.
Truss, J. K., Infinite permutation groups. II. Subgroups of small index, J. Algebra 120(2): 494–515, 1989.
Truss, J. K, and E. Vargas-García, Reconstructing the topology on monoids and polymorphism clones of reducts of the rationals, ArXiv e-prints, 2016.
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Presented by Daniele Mundici
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Pech, C., Pech, M. Reconstructing the Topology of the Elementary Self-embedding Monoids of Countable Saturated Structures. Stud Logica 106, 595–613 (2018). https://doi.org/10.1007/s11225-017-9756-6
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DOI: https://doi.org/10.1007/s11225-017-9756-6