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The essential arity of clones over algebras. (English) Zbl 1235.08002
Chajda, I. (ed.) et al., Proceedings of the 79th workshop on general algebra “79. Arbeitstagung Allgemeine Algebra”, 25th conference of young algebraists, Palacký University Olomouc, Olomouc, Czech Republic, February 12–14, 2010. Klagenfurt: Verlag Johannes Heyn (ISBN 978-3-7084-0407-3/pbk). Contributions to General Algebra 19, 139-149 (2010).
A clone \(C\) on a set \(A\) is a set of finitary operations on \(A\) that is closed under composition and contains all projections. By choosing an appropriate algebraic structure on \(A\), the elements in \(C\) can be viewed as morphisms in a quasivariety. D. Mašulović [Int. J. Algebra Comput. 16, No. 4, 657–687 (2006; Zbl 1108.06007)] proposed this approach to dualize clones on finite sets as categories. The dualized clone is then formed by morphisms from a finite relational structure to its finite copowers in the dual category.
In this paper it is shown that the structure of these copowers uniquely determines whether the essential arity of the operations in a clone is bounded (i.e., whether there exists a positive integer \(k\) such that every operation in the clone depends on at most \(k\) of its arguments).
For the entire collection see [Zbl 1201.08001].

08A40 Operations and polynomials in algebraic structures, primal algebras
08C20 Natural dualities for classes of algebras