Abstract
We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function \({f : \mathbb{N} \rightarrow \mathbb{N}}\) such that the n-ary part of C is equal to the n-ary part of Pol \({{\rm Inv}^{[f(n)]} C}\), where \({{\rm Inv}^{[f(n)]} C}\) denotes the set of f(n)-ary invariant relations of C.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aichinger E.: Local polynomial functions on the integers. Riv. Mat. Univ. Parma 6(5), 169–177 (1997)
Aichinger, E.: The structure of composition algebras. Ph.D. thesis, Johannes Kepler Universität Linz (1998). Available at www.algebra.uni-linz.ac.at/~erhard/Diss/
Burris, S., Sankappanavar, H.P.: A course in universal algebra. Springer New York Heidelberg Berlin (1981)
Eigenthaler G.: Einige Bemerkungen über Clones und interpolierbare Funktionen auf universellen Algebren. Beiträge Algebra Geom. 15, 121–127 (1983)
Goldstern M., Pinsker M.: A survey of clones on infinite sets. Algebra Universalis 59, 365–403 (2008)
Hall R.R.: On pseudo-polynomials. Mathematika 18, 71–77 (1971)
Hule H., Nöbauer W.: Local polynomial functions on universal algebras. Anais da Acad. Brasiliana de Ciencias 49, 365–372 (1977)
Kaarli K.: Affine complete abelian groups. Math.Nachr. 107, 235–239 (1982)
Pöschel, R.: A general Galois theory for operations and relations and concrete characterization of related algebraic structures, Report 1980, vol. 1. Akademie der Wissenschaften der DDR, Institut für Mathematik, Berlin (1980)
Pöschel, R., Kalužnin, L.A.: Funktionen- und Relationenalgebren, Mathematische Monographien, vol. 15. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)
Szendrei, Á.: Clones in universal algebra. Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 99. Presses de l’Université de Montréal, Montreal, QC (1986)
Acknowledgements
Open access funding provided by Johannes Kepler University Linz. The author thanks Mike Behrisch for furnishing information on the reference [9].
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by R. Poeschel.
Supported by the Austrian Science Fund (FWF): P24077 and P29931.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Aichinger, E. On the local closure of clones on countable sets. Algebra Univers. 78, 355–361 (2017). https://doi.org/10.1007/s00012-017-0465-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-017-0465-9