To each submonoid
P
P
of a group we associate a universal Toeplitz
C
∗
\mathrm {C}^*
-algebra
T
u
(
P
)
\mathcal {T}_u(P)
defined via generators and relations;
T
u
(
P
)
\mathcal {T}_u(P)
is a quotient of Li’s semigroup
C
∗
\mathrm {C}^*
-algebra
C
s
∗
(
P
)
\mathrm {C}^*_s(P)
and they are isomorphic iff
P
P
satisfies independence. We give a partial crossed product realization of
T
u
(
P
)
\mathcal {T}_u(P)
and show that several results known for
C
s
∗
(
P
)
\mathrm {C}^*_s(P)
when
P
P
satisfies independence are also valid for
T
u
(
P
)
\mathcal {T}_u(P)
when independence fails. At the level of the reduced semigroup
C
∗
\mathrm {C}^*
-algebra
T
λ
(
P
)
\mathcal {T}_\lambda (P)
, we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of
P
P
, generalizing a result of Li for monoids with trivial unit group. We characterize when the action of the group of units is topologically free, in which case a representation of
T
λ
(
P
)
\mathcal {T}_\lambda (P)
is faithful iff it is jointly proper. This yields a uniqueness theorem that generalizes and unifies several classical results. We provide a concrete presentation for the covariance algebra of the product system over
P
P
with one-dimensional fibers in terms of a new notion of foundation sets of constructible ideals. We show that the covariance algebra is a universal analogue of the boundary quotient and give conditions on
P
P
for the boundary quotient to be purely infinite simple. We discuss applications to a numerical semigroup and to the
a
x
+
b
ax+b
-monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails.