  The note is concerned with inductive systems of Toeplitz algebras and their
$*$-homomorphisms over arbitrary partially ordered sets. The Toeplitz algebra
is the reduced semigroup $C^*$-algebra for the additive semigroup of
non-negative integers. It is known that every partially ordered set can be
represented as the union of the family of its maximal upward directed subsets
indexed by elements of some set. In our previous work we have studied a
topology on this set of indexes. For every maximal upward directed subset we
consider the inductive system of Toeplitz algebras that is defined by a given
inductive system over an arbitrary partially ordered set and its inductive
limit. Then for a base neighbourhood $U_a$ of the topology on the set of
indexes we construct the $C^*$-algebra $\mathfrak{B}_a$ which is the direct
product of those inductive limits. In this note we continue studying the
connection between the properties of the topology on the set of indexes and
properties of inductive limits for systems consisting of $C^*$-algebras
$\mathfrak{B}_a$ and their $*$-homorphisms. It is proved that there exists an
embedding of the reduced semigroup $C^*$-algebra for a semigroup in the
additive group of all rational numbers into the inductive limit for the system
of $C^*$-algebras $\mathfrak{B}_a$.
