The cardinality of the set of all clones containing a given minimal clone on three elements
Let \(E_3=\{0,1,2\}\) and denote \(P_3^n\) the set of all \(n\)-ary functions on \(E_3\), let \(P_3=\bigcup_{ n \geq 1} P_3^n\). A set \(F \subseteq P_3\) is called a clone if \(F\) is closed under composition and contains all projections. Let \([F]\) denote the closure of a set \(F \subseteq P_3\) under composition, and \(J_3\) the set of all projections, then a clone \(A \neq J_3\) is called minimal if \(\forall f \in A \backslash J_3 [ \{f\} \cup J_3] = A\). In [Acta Cybern. 6, 227--238 (1983; Zbl 0537.08002)], \textit{B. Csákány} showed that \(P_3\) has exactly 84 minimal clones. For a good survey on minimal clones we refer to a minicourse by \textit{B. Csákány} [Algebra Univers. 54, No. 1, 73--89 (2005; Zbl 1088.08002)]. The current paper is devoted to the cardinality of the set of all clones containing a given minimal clone of \(P_3\). The cardinality for many minimal clones was considered by \textit{J. Pantović} and \textit{D. Vojvodić} [Mult.-Valued Log. 5, No. 5, 367--371 (2000; Zbl 0993.08006); Novi Sad J. Math. 30, No. 2, 155--158 (2000; Zbl 1265.08008); Acta Cybern. 14, No. 3, 491--495 (2000; Zbl 0949.08002)], \textit{I. G. Rosenberg} [Colloq. Math. Soc. János Bolyai 43, 405--427 (1986; Zbl 0603.08005)] and S. S. Marchenkov. In this paper, the remaining cases are considered.
scientific article; zbMATH DE number 6132827
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