  Algebraic Petri nets are a formalism for modeling distributed systems and
algorithms, describing control and data flow by combining Petri nets and
algebraic specification. One way to specify correctness of an algebraic Petri
net model $N$ is to specify a linear equation $E$ over the places of $N$ based
on term substitution, and coefficients from an abelian group $G$. Then, $E$ is
valid in $N$ iff $E$ is valid in each reachable marking of $N$ . Due to the
expressive power of Algebraic Petri nets, validity is generally undecidable.
Stable linear equations form a class of linear equations for which validity is
decidable. Place invariants yield a well-understood but incomplete
characterization of all stable linear equations. In this paper, we provide a
complete characterization of stability for the subclass of homogeneous linear
equations, by restricting ourselves to the interpretation of terms over the
Herbrand structure without considering further equality axioms. Based thereon,
we show that stability is decidable for homogeneous linear equations if $G$ is
a cyclic group.
