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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 copies is still a Cantorval for any . We also prove that for any , , the algebraic sum of copies of a Cantor set can transit from a Cantor set to a Cantorval for and then to an interval for . 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.