In $[4]$, Baker and Rejali are concerned with the regularity of
$\ell_1(S, w)$, where $S$ is a discrete semigroup. We study
the Arens regularity of the weighted semigroup algebra $M_b(S, w)$,
for non-discrete $S$. We show that $\ell_1(S, w)$ is regular
if and only if $M_b(S, w)$ is regular.
We obtain conditions for the regularity of $M^{\ell}_a(S, w)$,
analogous to the weighted group algebra $L^1(G, w)$. It is shown
that $L^1(G, w)$ is regular if and only if $G$ is finite or $G$
is discrte and $\O$ is $0-$cluster, where $\O(x,y)=w(xy)/w(x)w(y)$
for $x,y \in G$. We obtain that $M^{\ell}_a(S,w)$ is regular
whenever $M^{\ell}_a(S)$ is. Furthermore, $M^{\ell}_a(S,w)$ is
regular if and only if $\ell_1(F_a(S), w)$ is regular,
where $F_a(S)$ is the foundation semigroup of $M^{\ell}_a(S,w)$.
For a foundation semigroup $S$, the regularity of
$M_b(S,w), M^{\ell}_a(S,w)$ and $\ell_1(S, w)$ are equivalent.