  We study the Toeplitz algebra $\TT(\N\rtimes\N^\times)$ and three quotients
of this algebra: the $C^*$-algebra $\qn$ recntly introduced by Cuntz, and two
new ones, which we call the additive and multiplicative boundary quotients.
These quotients are universal for Nica-covariant representations of
$\N\rtimes\N^\times$ satisfying extra relations, and can be realised as partial
crossed products. We use the structure theory for partial crossed products to
prove a uniqueness theorem for the additive boundary quotient, and use the
recent analysis of KMS states on $\TT(\nxnx)$ to describe the KMS states on the
two quotients. We then show that $\TT(\nxnx)$, $\qn$ and our new quotients are
all interesting new examples for Larsen's theory of Exel crossed products by
semigroups.
