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Finite Clones Containing All Permutations

Published online by Cambridge University Press:  20 November 2018

L. Haddad  and
I. G. Rosenberg
L. Haddad
Affiliation:
Department of Mathematics and Computer Science RMC Kingston, Ontario K7K5L0
I. G. Rosenberg
Affiliation:
Mathématiqués et Statistics, Université de Montréal C.R 6128 Succ., Centre-ville Montréal, Québec H3C 3J7
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Abstract

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Let A be a finite set with |A| > 2. We describe all clones on A containing the set SA of all permutations of A among its unary operations. (A clone on A is a composition closed set of finitary operations on A containing all projections). With a few exceptions such a clone C is either essentially unary or cellular i.e. there exists a monoid M of self-maps of A containing SA such that either C = (= all essentially unary operations agreeing with some f ∊ M) or C = ∪ Гh where 1 < h ≤ |A| and Гh consists of all finitary operations on A taking at most h values. The exceptions are subclones of Burle's clone or of its variant (provided |A| is even).

Keywords

08A0508A4008A02
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 951 - 970
Copyright
Copyright © Canadian Mathematical Society 1994

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