  To each monoid $P$ that embeds in a group we associate a universal Toeplitz
C*-algebra $T_u(P)$ defined via generators and relations; $T_u(P)$ is a
quotient of Li's semigroup C*-algebra $C^*(P)$ and they are isomorphic if and
only if $P$ satisfies independence. We give a partial crossed product
realization of $T_u(P)$ and show that several key results known for $C^*(P)$
when $P$ satisfies independence are also valid for $T_u(P)$ when independence
fails. At the level of the reduced semigroup C*-algebra $T_r(P)$, 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 $P$,
generalizing a result of Li for monoids with trivial unit group. We also
characterize when the action of the group of units is topologically free and we
show that in this case a representation of $T_r(P)$ is faithful iff it is
jointly proper. This yields a uniqueness theorem for C*-algebras generated by
semigroups of isometries that unifies several classical results. We provide a
presentation for the covariance algebra of the product system over $P$ with
one-dimensional fibers in terms of a notion of foundation sets of constructible
ideals that generalizes that of Sims and Yeend for quasi-lattice orders. The
covariance algebra is a full, or universal, analogue of the boundary quotient.
We give purely algebraic sufficient conditions on $P$ for the boundary quotient
to be purely infinite simple, which reduce to Starling's conditions in the case
of right LCM monoids. We discuss applications of our results to examples that
include a numerical semigroup and the $ax+b$-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. In addition, we simplify and
generalize results for right LCM monoids.
