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A maximal clone of monotone operations which is not finitely generated. (English) Zbl 0614.08006

A set of finitary operations on a finite set A is called a clone on A if it contains all projections and is closed under superposition. In his paper [Rozpr. Česk. Akad. Věd., Řada Mat. Přír. Věd. 80, No.4, 3-93 (1970; Zbl 0199.302)], I. G. Rosenberg classified the maximal clones. In the paper [Z. Math. Logik Grundl. Math. 24, 79-96 (1978; Zbl 0401.03008)] by D. Lau, it has been shown that for \(| A| \leq 7\) every maximal clone on A is finitely generated.
In the present paper, the author studies a maximal clone on a set of eight elements and proves, that this clone - which consists of all finitary operations preserving a certain partial order - is not finitely generated. Since it is already known that all other maximal clones on an eight element set are finitely generated, this result settles the case \(| A| =8\).
Reviewer: G.Eigenthaler

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
06A06 Partial orders, general
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References:

[1] K. A. Baker and A. F. Pixley (1975) Polynomial interpolations and the Chinese remainder theorem for algebraic systems, Math. Z. 143, 165-174. · Zbl 0298.08004 · doi:10.1007/BF01187059
[2] J. Demetrovics, L. Hann?k, and L. Ronyai (1984) Near unanimity functions and partial orderings, Proc. 14, ISMVL., Manitoba, pp. 52-56.
[3] D. Lau (1978) Bestimmung der Ordnung maximaler Klassen von Funktionen der k-wertigen Logik. Z. Math. Logik u. Grundl. Math. 24, 79-96. · Zbl 0401.03008 · doi:10.1002/malq.19780240111
[4] R. P?schel and L. A. Kalu?nin (1979) Funktionen-und Relationenalgebren, VEB Deutscher Verlag der Wissenschaften, Berlin.
[5] I. G. Rosenberg (1970) ?ber die funktionale Vollstandigkeit in den mehrwertigen Logiken, Rozpr. ?SAV ?ada Mat. P?ir. V?d. 80, 4, 3-93.
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