Unary operations with long pre-periods (Q947823)

For a unary operation \(f: A\to A\) defined on a finite set \(A\), there is a least natural number \(m\) such that \(f^m[A]=f^{m+1}[A]\) (where \(f^k[A]\) is the image of \(A\) under \(k\) applications of \(f\)). The authors call this number the pre-period of \(f\). For \(A\) of size \(n\), those \(f\) which have pre-period \(n-1\) and \(n-2\), respectively, are computed. Then those equivalence relations which are congruence relations with respect to an operation that has such a ``long'' pre-period (i.e., \(n-2\) or \(n-1\)) are calculated. There are not many such equivalence relations, and they can be found by distinguishing cases and elementary calculations. As a corollary, the authors state that every affine complete non-simple algebra on a base set \(A\) of at most \(5\) elements has a polynomial operation with pre-period \(n-2\) or \(n-1\). Moreover, they state that any group that has a polynomial operation with pre-period \(n-1\) is simple, and that every non-trivial normal subgroup of a group with a polynomial operation with pre-period \(n-2\) has index 2.