Abstract
We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that: For example, the lattice \(M_{2^\omega}\) is a sublattice of the local clone lattice.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bodirsky, M., Chen, H., Pinsker, M.: The reducts of equality up to primitive positive interdefinability. J. Symb. Log. 75(4), 1249–1292 (2010). Preprint available from http://arxiv.org/abs/0810.2270
Bulatov, A., Krokhin, A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (2005)
Bodirsky, M., Nešetřil, J.: Constraint satisfaction with countable homogeneous templates. J. Log. Comput. 16(3), 359–373 (2006)
Bodirsky, M.: Constraint satisfaction with infinite domains. Dissertation, Humboldt-Universität zu Berlin (2004)
Bulatov, A.: Identities in lattices of closed classes. Discrete Math. Appl. 3(6), 601–609 (1993)
Bulatov, A.: Finite sublattices in the lattice of clones. Algebra Logic 33(5), 287–306 (1994)
Cameron, P.J.: Transitivity of permutation groups on unordered sets. Math. Z. 148, 127–139 (1976)
Cameron, P.J.: Oligomorphic permutation groups. London Mathematical Society Lecture Note Series, vol. 152. Cambridge University Press, Cambridge (1990)
Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice-Hall (1973)
Goldstern, M., Pinsker, M.: A survey of clones on infinite sets. Algebra Univers. 59, 365–403 (2008)
Junker, M., Ziegler, M.: The 116 reducts of (ℚ, < , a). J. Symb. Log. 73(3), 861–884 (2008)
Pinsker, M.: Algebraic lattices are complete sublattices of the clone lattice on an infinite set. Fundam. Math. 195(1), 1–10 (2007)
Pinsker, M.: Sublattices of the lattice of local clones. In: Proceedings of the ROGICS’08 Conference, pp. 80–87 (2008)
Romov, B.A.: Galois correspondence between iterative post algebras and relations on infinite sets. Cybernetics 3, 377–379 (1977)
Rosenberg, I.G., Schweigert, D.: Locally maximal clones. Elektron. Informationsverarb. Kybernet. 18(7–8), 389–401 (1982) (in German, Russian summary)
Rosenberg, I.G., Szabó, L.: Local completeness. I. Algebra Univers. 18(3), 308–326 (1984)
Szendrei, Á.: Clones in universal algebra. Les Presses de L’Université de Montréal (1986)
Thomas, S.: Reducts of the random graph. J. Symb. Log. 56(1), 176–181 (1991)
Tůma, J.: Intervals in subgroup lattices of infinite groups. J. Algebra 125(2), 367–399 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is grateful for support through Erwin Schrödinger Fellowship J2742-N18 of the Austrian Science Fund (FWF).
Rights and permissions
About this article
Cite this article
Pinsker, M. More Sublattices of the Lattice of Local Clones. Order 27, 353–364 (2010). https://doi.org/10.1007/s11083-010-9179-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-010-9179-8