Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T22:36:49.844Z Has data issue: false hasContentIssue false
  • English
  • Français

A universality result for endomorphism monoids of some ultrahomogeneous structures

Part of: General theory of categories and functors Semigroups Algebraic structures Model theory

Published online by Cambridge University Press:  16 March 2012

Igor Dolinka  and
Dragan Mašulović
Igor Dolinka
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia ([email protected]; [email protected])
Dragan Mašulović
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia ([email protected]; [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We devise a fairly general sufficient condition ensuring that the endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a Fraïssé limit) embeds all countable semigroups. This approach not only provides us with a framework unifying the previous scattered results in this vein, but actually yields new applications for endomorphism monoids of the (rational) Urysohn space and the countable universal ultrahomogeneous semilattice.

Keywords

endomorphism monoidFraïssé limitpushout

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 03C15: Denumerable structures 08A35: Automorphisms, endomorphisms 18A30: Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 20M50: Connections of semigroups with homological algebra and category theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 635 - 656
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Albert, M. H. and Burris, S. N., Finite axiomatizations for existentially closed posets and semilattices, Order 3 (1986), 169–178.CrossRefGoogle Scholar
2.Bonato, A., Delić, D. and Dolinka, I., All countable monoids embed into the monoid of the infinite random graph, Discrete Math. 310 (2010), 373–375.CrossRefGoogle Scholar
3.Cameron, P. J., The random graph, in Algorithms and combinatorics, Volume 14 (ed. Graham, R. L. and Nešetřil, J.), pp. 333–351 (Springer, 1997).Google Scholar
4.Cameron, P. J., The random graph revisited, in European Congress of Mathematics, Barcelona, 10–14 July 2000, Volume I, Progess in Mathematics, Volume 201, pp. 267–274 (Birkhäuser, 2001).CrossRefGoogle Scholar
5.Cameron, P. J. and Nešetřil, J., Homomorphism-homogeneous relational structures, Combinator. Prob. Comput. 15 (2006), 91–103.CrossRefGoogle Scholar
6.Dolinka, I., The endomorphism monoid of the random poset contains all countable semigroups, Alg. Univers. 56 (2007), 469–474.CrossRefGoogle Scholar
7.Dolinka, I., The Bergman property for endomorphism monoids of some Fraïssé limits, Forum Math., doi:10.1515/FORM.2011.153 (in press).Google Scholar
8.Dolinka, I. and Mašulović, D., Remarks on homomorphism-homogeneous lattices and semilattices, Monatsh. Math. 164 (2011), 23–37.CrossRefGoogle Scholar
9.Droste, M. and Kuske, D., On random relational structures, J. Combin. Theory A 102 (2003), 241–254.CrossRefGoogle Scholar
10.Droste, M., Kuske, D. and Truss, J. K., On homogeneous semilattices and their automorphism groups, Order 16 (1999), 31–56.CrossRefGoogle Scholar
11.Droste, M. and Macpherson, H. D., The automorphism group of the universal distributive lattice, Alg. Univers. 43 (2000), 295–306.CrossRefGoogle Scholar
12.Fraïssé, R., Sur certains relations qui généralisent l'ordre des nombres rationnels, C. R. Acad. Sci. Paris 237 (1953), 540–542.Google Scholar
13.Fraïssé, R., Sur l'extension aux relations de quelques propriétés des ordres, Annales Sci. École Norm. Sup. 71 (1954), 363–388.CrossRefGoogle Scholar
14.Grätzer, G., Universal algebra, 2nd edn (Springer, 2008).Google Scholar
15.Henson, C. W., A family of countable homogeneous graphs, Pac. J. Math. 38 (1971), 69–83.CrossRefGoogle Scholar
16.Hodges, W., A shorter model theory (Cambridge University Press, 1997).Google Scholar
17.Hubička, J. and Nešetřil, J., A finite presentation of the rational Urysohn space, Topol. Applic. 155 (2008), 1483–1492.CrossRefGoogle Scholar
18.Katětov, M., On universal metric spaces, in Proc. 6th Prague Topological Symp., Prague, 1986 (ed. Frolík, Z.), pp. 323–330 (Heldermann, Berlin, 1988).Google Scholar
19.Lachlan, A. H. and Woodrow, R. E., Countable ultrahomogeneous undirected graphs, Trans. Am. Math. Soc. 262 (1980), 51–94.CrossRefGoogle Scholar
20.Lane, S. Mac, Categories for the working mathematician, 2nd edn (Springer, 1998).Google Scholar
21.Mudrinski, N., Notes on endomorphisms of Henson graphs and their complements, Ars Combin. 96 (2010), 173–183.Google Scholar
22.Pestov, V., The isometry group of the Urysohn space as a Lévy group, Topol. Applic. 154 (2007), 2173–2184.CrossRefGoogle Scholar
23.Schmerl, J. H., Countable homogeneous partially ordered sets, Alg. Univers. 9 (1979), 317–321.CrossRefGoogle Scholar
24.Urysohn, P., Sur un espace métrique universel, Bull. Math. Sci. 51 (1927), 43–64, 74–96.Google Scholar
25.Uspenskij, V. V., On the group of isometries of the Urysohn universal metric space, Commentat. Math. Univ. Carolinae 31 (1990), 181–182.Google Scholar
26.Uspenskij, V. V., The Urysohn universal metric space is homeomorphic to a Hilbert space, Topol. Applic. 139 (2004), 145–149.CrossRefGoogle Scholar
27.Vershik, A. M., A random metric space is a Urysohn space, Dokl. Akad. Nauk SSSR 387 (2002), 733–736 (in Russian).Google Scholar