Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T00:49:13.581Z Has data issue: false hasContentIssue false
  • English
  • Français

PROJECTIVE CLONE HOMOMORPHISMS

Part of:Algebraic structuresModel theory

Published online by Cambridge University Press:  03 May 2019

MANUEL BODIRSKY,
MICHAEL PINSKER and
ANDRÁS PONGRÁCZ
MANUEL BODIRSKY
Affiliation:
INSTITUT FÜR ALGEBRA TECHNISCHE UNIVERSITÄT DRESDEN 01062 DRESDEN, GERMANY E-mail: Manuel.Bodirsky@tu-dresden.de URL: http://www.math.tu-dresden.de/~bodirsky/
MICHAEL PINSKER
Affiliation:
INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE FG ALGEBRA, TECHNISCHE UNIVERSITÄT WIEN WIEN, AUSTRIA and DEPARTMENT OF ALGEBRA CHARLES UNIVERSITY PRAGUE, CZECH REPUBLIC E-mail: marula@gmx.at  URL: http://dmg.tuwien.ac.at/pinsker/
ANDRÁS PONGRÁCZ
Affiliation:
DEPARTMENT OF ALGEBRA AND NUMBER THEORY UNIVERSITY OF DEBRECEN 4032 DEBRECEN, EGYETEM SQUARE 1, HUNGARY E-mail: pongracz.andras@science.unideb.hu
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.

It is known that a countable $\omega $-categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.

Keywords

clonehomogeneous structureautomatic continuitytrivial algebrapolymorphism cloneconstraint satisfaction problemcanonical function

MSC classification

Primary:03C05: Equational classes, universal algebra
Secondary:03C40: Interpolation, preservation, definability08A70: Applications of universal algebra in computer science08A35: Automorphisms, endomorphisms08A30: Subalgebras, congruence relations

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021, pp. 148 - 161
Check for updates
Copyright
© The Association for Symbolic Logic 2019

References

REFERENCES

Barto, L., Kompatscher, M., Olšák, M., Van Pham, T., and Pinsker, M., The equivalence of two dichotomy conjectures for infinite domain constraint satisfaction problems, Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science—LICS’17, 2017.10.1109/LICS.2017.8005128CrossRefGoogle Scholar
Barto, L., Kompatscher, M., Olšák, M., Van Pham, T., and Pinsker, M., Equations in oligomorphic clones and the constraint satisfaction problem for omega-categorical structures. Journal of Mathematical Logic , vol. 19 (2019), no. 2, article no. 1950010.Google Scholar
Barto, L., Opršal, J., and Pinsker, M., The wonderland of reflections . Israel Journal of Mathematics , vol. 223 (2018), no. 1, pp. 363–398.CrossRefGoogle Scholar
Barto, L. and Pinsker, M., The algebraic dichotomy conjecture for infinite domain constraint satisfaction problems , Proceedings of the 31th Annual IEEE Symposium on Logic in Computer Science—LICS’16 , 2016, pp. 615–622. Preprint available from arXiv:1602.04353.Google Scholar
Barto, L. and Pinsker, M., Topology is irrelevant. SIAM Journal on Computing , vol. 49 (2020), no. 2, pp. 365–393.CrossRefGoogle Scholar
Birkhoff, G., On the structure of abstract algebras . Mathematical Proceedings of the Cambridge Philosophical Society , vol. 31 (1935), no. 4, pp. 433–454.CrossRefGoogle Scholar
Bodirsky, M., Cores of countably categorical structures . Logical Methods in Computer Science , vol. 3 (2007), no. 1, pp. 1–16.CrossRefGoogle Scholar
Bodirsky, M. and Mottet, A., A dichotomy for first-order reducts of unary structures . Logical Methods in Computer Science , vol. 14 (2018), no. 2, pp. 1–31.Google Scholar
Bodirsky, M. and Pinsker, M., Reducts of Ramsey structures . AMS Contemporary Mathematics , vol. 558 (2011) (Model theoretic methods in finite combinatorics), pp. 489–519.CrossRefGoogle Scholar
Bodirsky, M. and Pinsker, M., Minimal functions on the random graph . Israel Journal of Mathematics , vol. 200 (2014), no. 1, pp. 251–296.CrossRefGoogle Scholar
Bodirsky, M. and Pinsker, M., Schaefer’s theorem for graphs . Journal of the Association for Computing Machinery , vol. 62 (2015), no. 3, Article no. 19, pp. 52–96. A conference version appeared in the Proceedings of STOC 2011, pp. 655–664.CrossRefGoogle Scholar
Bodirsky, M. and Pinsker, M., Topological Birkhoff . Transactions of the American Mathematical Society , vol. 367 (2015), pp. 2527–2549.10.1090/S0002-9947-2014-05975-8CrossRefGoogle Scholar
Bodirsky, M. and Pinsker, M., Canonical functions: A proof via topological dynamics. Contributions to Discrete Mathematics , to appear. arXiv:1610.09660.Google Scholar
Bodirsky, M., Hils, M., and Martin, B., On the scope of the universal-algebraic approach to constraint satisfaction. Proceedings of the Annual Symposium on Logic in Computer Science (LICS) , IEEE Computer Society, July 2010, pp. 90–99.CrossRefGoogle Scholar
Bodirsky, M., Madelaine, F., and Mottet, A., A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP, Proceedings of the Symposium on Logic in Computer Science—LICS’18, 2018. Preprint available from arXiv:1802.03255.10.1145/3209108.3209156CrossRefGoogle Scholar
Bodirsky, M., Pinsker, M., and Pongrácz, A., Reconstructing the topology of clones . Transactions of the American Mathematical Society , vol. 369 (2017), pp. 3707–3740.CrossRefGoogle Scholar
Bodirsky, M., Pinsker, M., and Tsankov, T., Decidability of definability , this Journal, vol. 78 (2013), no. 4, pp. 1036–1054. A conference version appeared in the Proceedings of the Twenty-Sixth Annual IEEE Symposium on Logic in Computer Science, LICS 2011, pp. 321–328.Google Scholar
Bodirsky, M., Martin, B., Pinsker, M., and Pongrácz, A., Constraint satisfaction problems for reducts of homogeneous graphs. SIAM Journal on Computing , vol. 48 (2019), no. 4, pp. 1224–1264.10.1137/16M1082974CrossRefGoogle Scholar
Bodirsky, M., Mottet, A., Olšák, M., Opršal, J., Pinsker, M., and Willard, R., Topology is relevant (in the infinite-domain dichotomy conjecture for constraint satisfaction problems), Proceedings of the Symposium on Logic in Computer Science—LICS’19 , 2019. Preprint available from arXiv:1901.04237.CrossRefGoogle Scholar
Bulatov, A. A., A dichotomy theorem for nonuniform CSPs , 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15–17, 2017 , 2017, pp. 319–330.Google Scholar
Gehrke, M. and Pinsker, M., Uniform Birkhoff . Journal of Pure and Applied Algebra , vol. 222 (2018), no. 5, pp. 1242–1250.10.1016/j.jpaa.2017.06.016CrossRefGoogle Scholar
Hodges, W., A Shorter Model Theory , Cambridge University Press, Cambridge, 1997.Google Scholar
Lascar, D., Autour de la propriété du petit indice . Proceedings of the London Mathematical Society , vol. 62 (1991), no. 1, pp. 25–53.CrossRefGoogle Scholar
Post, E. L., The Two-Valued Iterative Systems of Mathematical Logic , Annals of Mathematics Studies, vol. 5, Princeton University Press, Princeton, NJ, 1941.Google Scholar
Siggers, M. H., A strong Mal’cev condition for varieties omitting the unary type . Algebra Universalis , vol. 64 (2010), no. 1, pp. 15–20.CrossRefGoogle Scholar
Zhuk, D., A proof of CSP dichotomy conjecture , 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017 , 2017, pp. 331–342.CrossRefGoogle Scholar