Congruence preserving functions on free monoids (Q1686332)

Let \((A,\Omega)\) be an algebra and let \(k\) be a positive integer. A function \(f\colon A^k\rightarrow A\) is \textit{congruence preserving} if every congruence of the algebra \((A,\Omega)\) is also a congruence of the expanded algebra \((A,\Omega\cup\left\{f\right\})\). An algebra is \textit{affine complete} if each congruence preserving function is a \textit{polynomial function}, i.e. belongs to the smallest clone of operations containing \(\Omega\) and all constant functions. The main result of the paper shows that the free monoid on at least 3 generators is affine complete. To obtain this the characterization of all congruence preserving functions for such monoids is given. In [\textit{P. Cégielski} et al., Int. J. Number Theory 11, No. 7, 2109--2139 (2015; Zbl 1395.11006)], it was shown that for the free monoid on one generator there exist non-polynomial congruence preserving functions. The open problem is whether the free monoid on 2 generators is affine complete.