On the number of operations in a clone

Authors:
Joel Berman and Andrzej Kisielewicz

Journal:
Proc. Amer. Math. Soc. **122** (1994), 359-369

MSC:
Primary 08A40

DOI:
https://doi.org/10.1090/S0002-9939-1994-1198450-9

MathSciNet review:
1198450

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Abstract: A clone *C* on a set *A* is a set of operations on *A* containing the projection operations and closed under composition. A combinatorial invariant of a clone is its -sequence , where is the number of essentially *n*-ary operations in *C*. We investigate the links between this invariant and structural properties of clones. It has been conjectured that the -sequence of a clone on a finite set is either eventually strictly increasing or is bounded above by a finite constant. We verify this conjecture for a large family of clones. A special role in our work is played by totally symmetric operations and totally symmetric clones. We show that every totally symmetric clone on a finite set has a bounded -sequence and that it is decidable if a clone is totally symmetric.

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1198450-9

Article copyright:
© Copyright 1994
American Mathematical Society