  For a locally compact group $G$, the measure convolution algebra $M(G)$
carries a natural coproduct. In previous work, we showed that the canonical
predual $C_0(G)$ of $M(G)$ is the unique predual which makes both the product
and the coproduct on $M(G)$ weak$^*$-continuous. Given a discrete semigroup
$S$, the convolution algebra $\ell^1(S)$ also carries a coproduct. In this
paper we examine preduals for $\ell^1(S)$ making both the product and the
coproduct weak$^*$-continuous. Under certain conditions on $S$, we show that
$\ell^1(S)$ has a unique such predual. Such $S$ include the free semigroup on
finitely many generators. In general, however, this need not be the case even
for quite simple semigroups and we construct uncountably many such preduals on
$\ell^1(S)$ when $S$ is either $\mathbb Z_+\times\mathbb Z$ or $(\mathbb
N,\cdot)$.
