Summary
A clone is a set of composition closed functions on some set. A non-trivial fact is that on a finite set every clone contains a minimal clone. This naturally leads to the problem of classifying all minimal clones on a finite set. In this paper I survey what is known about this classification. Rather than repeat the arguments used in the original papers, I have tried to use known results about finite algebras to give a more coherent and unified description of the known minimal clones.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Buekenhout, F., Delandtsheer, A., Doyen, J., Kleidman, P., Liebeck, M. and Saxl, J., Linear spaces with flag transitive automorphisms groups. Geom. Dedicata 36 (1990), 89–94.
Csákány B., All minimal clones on a three-element set. Acta Cybernet. 6 (1983),227–238.
Csakany, B., On conservative minimal operations. [Coll. Math. Soc. János Bolyai, No. 43]. North-Holland, Amsterdam, 1986, 49–60.
Ježek, J. and Quackenbush, R., Directoids: algebraic models of up-directed sets. Algebra Univ. 27, (1990) 49–69.
Ježek, J. and Quackenbush, R., Minimal clones of conservative functions. Preprint.
Mckenzie, R., Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties. Algebra Univ. 8 (1978), 336–348.
Padmanabhan, R. and Wolk, B., Equational theories with a minority polynomial. Proc. Amer. Math. Soc. 83 (1981), 238–242.
Paley P. P., The arity of minimal clones. Acta Sci. Math. (Szeged) 50 (1986), 331–333.
Palfy P.P., Minimal clones. Manuscript.
Pöschel, R. and Kauluynin, L. A., Funktionen-und Relationenalgebren. Deutscher Verl. Wiss., Berlin, 1979.
Rosenberg, I. G., Minimal clones I: the five types. [Coll. Math. Soc. Janos Bolyai, No. 43]. North-Holland, Amsterdam, 1986, 405–427.
Świerczkowski, S., Algebras which are independently generated by every n elements. Fund. Math. 49 (1960), 93–104.
Szendrei, Á, Clones in universal algebra. [Sém. de Math. Supp., No. 99]. Les Presses de l’Université de Montréal, Montréal, 1986.
Szendrei, Á, Idempotent algebras with restrictions on subalgebras. Acta Sci. Math. (Szeged) 51 (1987), 251–268.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Trevor Evans
Rights and permissions
Copyright information
© 1995 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Quackenbush, R.W. (1995). A survey of minimal clones. In: Aczél, J. (eds) Aggregating clones, colors, equations, iterates, numbers, and tiles. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9096-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9096-0_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-5243-1
Online ISBN: 978-3-0348-9096-0
eBook Packages: Springer Book Archive