Binary operations suffice to test collapsing of monoidal intervals
Suppose that \(M\) is a transformation monoid on a set \(A\) and let \({\langle }M{\rangle }\) be the set of all \(n\)-ary functions \(f\) over \(A\) such that for some \(m\in M\) and \(1\leq i \leq n\) \(f(x_1,\ldots ,x_n)\approx m(x_i)\) holds. \(M\) is called collapsing if \({\langle }M{\rangle }= \text{Sta} M\), the stabilizer of \(M\). \textit{P. P. Pálfy} proved that for every non-collapsing monoid \(M\) there is a 5-ary function \(f\in \text{Sta} M\setminus {\langle }M{\rangle }\). The author proves the following generalization of Pálfy's theorem: Let \(| A | \geq 3\). Let \(M\) be a non-collapsing monoid on \(A\). Then \(\text{Sta}M\) contains an essentially binary function. In other words, the stabilizer clone \(\text{Sta} M\) is either trivial or contains essentially binary function.
scientific article; zbMATH DE number 1226227
scientific article