  In this paper for a finite field $F$, a nonempty set $\Gamma$, a self--map
$\varphi:\Gamma\to\Gamma$ and a weight vector $\mathfrak{w}\in F^\Gamma$, we
show that the set--theoretical entropy of the weighted generalized shift
$\sigma_{\varphi,\mathfrak{w}}:F^\Gamma\to F^\Gamma$ is either zero or
$+\infty$, moreover it is equal to zero if and only if
$\sigma_{\varphi,\mathfrak{w}}$ is quasi--periodic. On the other hand after
characterizing all conditions under which
$\sigma_{\varphi,\mathfrak{w}}:F^\Gamma\to F^\Gamma$ is of finite fibre, we
show that the cotravariant set--theoretical entropy of the finite fibre
$\sigma_{\varphi,\mathfrak{w}}:F^\Gamma\to F^\Gamma$ depends only on $\varphi$
and $\rm{supp}(\mathfrak{w})$. In final sections we study the restriction of
$\sigma_{\varphi,\mathfrak{w}}$ to the direct sum
$\mathop{\bigoplus}\limits_{\Gamma}F$.
