  Let $E$ be the self-similar set generated by the {\it iterated function
system} {\[ f_0(x)=\frac{x}{\beta},\quad f_1(x)=\frac{x+1}{\beta}, \quad
f_{\beta+1}=\frac{x+\beta+1}{\beta} \]}with $\beta\ge 3$. {Then} $E$ is a
self-similar set with complete {overlaps}, i.e., $f_{0}\circ
f_{\beta+1}=f_{1}\circ f_1$, but $E$ is not totally self-similar.
  We investigate all its generating iterated function systems, give the
spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of
$E$ and of the sets which contain all points in $E$ having finite or infinite
different triadic codings.
