Title: Arens regularity of certain weighted semigroup algebras and countability
Abstract: In this paper, among other things, we show that for a large class of semigroups, the Arens regularity of the weighted semigroup algebra $$\ell _1(S,\omega )$$ implies the countability of S. This generalizes the main result of Craw and Young (Q J Math 25:351–358, 1974), stating that every countable semigroup admits a weight $$\omega $$ for which $$\ell _1(S,\omega )$$ is Arens regular and no uncountable group admits such a weight.