Arens regularity of weighted convolution algebras that arise from totally ordered semilattices

We study Arens regularity of weighted semigroup convolution algebras for the specific case of totally ordered semilattices. This paper is a natural continuation of that of Dales and Strauss (2022) [DS22], where they studied the unweighted case. We also study some other properties of these algebras, such as the existence of Banach-algebra preduals.