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.