Algebraic structures within subsets of Hamel and Sierpiński-Zygmund functions

Abstract

We prove the existence of an additive semigroup of cardinality $2^\mathfrak c$ contained in the intersection of the classes of Hamel functions ($\rm HF$) and Sierpiński-Zygmund functions ($\rm SZ$). In addition, we show that under certain set-theoretic assumptions the lineability of the class of Sierpiński-Zygmund functions ($\rm SZ$) is equal to the lineability of the class of almost continuous Sierpiński-Zygmund functions ($\rm AC\cap\rm SZ$).