Given a left module \(V\) over a not necessarily commutative field \(K\) the authors introduce the structure of the so-called symmetric groupoid on a subset of \(V\). Then they extend the natural notion of nonnegative powers \(x^{\langle k \rangle}\) of the elements \(x\) of this groupoid to the whole \(V\). Assuming additionally that \(V\) is normed and \(K\) is valuated they study iteration processes on \(V\) of the type \(x \mapsto x^{\langle k \rangle} + c\) and analogons of the Mandelbrot set appearing here in a natural way. A special case of complex normed linear spaces is also described.