On the maximal number of non-\(C\)-endorigid equivalence relations (Q1910732)

Let \(M\) denote a fixed set. A finitary function \(f\) and a finitary relation \(R\) on \(M\) are said to be mutually compatible if \(R\) is a subalgebra of a suitable direct power of \((M,f)\). A set \(E\) of equivalence relations on \(M\) is called \(C\)-endorigid if the identity function and the constant functions are the only unary functions on \(M\) being compatible with every element of \(E\). Let \(\mu(n)\) denote the maximal cardinality of a non-\(C\)-endorigid set of equivalence relations on an \(n\)-element set. \textit{H. Länger} and \textit{R. Pöschel} [ibid. 32, 129-142 (1984; Zbl 0558.08004)] proved that \(\mu(n) \geq 2\text{eq} (n - 1)\) if \(n\) is an integer with \(n \geq 3\) (here \(\text{eq}(m)\) denotes the number of equivalence relations on an \(m\)-element set) and the determination of the exact value of \(\mu(n)\) was posed there as an open problem. In the paper under review it is proved that in fact \(\mu(n) = 2\text{eq}(n - 1)\) if \(n\) is an integer with \(n \geq 3\).