  In this paper we look at the topological type of algebraic sum of achievement
sets. We show that there is a Cantorval such that the algebraic sum of its $k$
copies is still a Cantorval for any $k \in \mathbb{N}$. We also prove that for
any $m,p \in (\mathbb{N}\setminus \{1\}) \cup \{\infty\}$, $p \geq m$, the
algebraic sum of $k$ copies of a Cantor set can transit from a Cantor set to a
Cantorval for $k=m$ and then to an interval for $k=p$. These two main results
are based on a new characterization of sequences whose achievement sets are
Cantorvals. We also define a new family of achievable Cantorvals which are not
generated by multigeometric series. In the final section we discuss various
decompositions of sequences related to the topological typology of achievement
sets.
