  For models of concurrent and distributed systems, it is important and also
challenging to establish correctness in terms of safety and/or liveness
properties. Theories of distributed systems consider equivalences fundamental,
since they (1) preserve desirable correctness characteristics and (2) often
allow for component substitution making compositional reasoning feasible.
Modeling distributed systems often requires abstraction utilizing
nondeterminism which induces unintended behaviors in terms of infinite
executions with one nondeterministic choice being recurrently resolved, each
time neglecting a single alternative. These situations are considered
unrealistic or highly improbable. Fairness assumptions are commonly used to
filter system behaviors, thereby distinguishing between realistic and
unrealistic executions. This allows for key arguments in correctness proofs of
distributed systems, which would not be possible otherwise. Our contribution is
an equivalence spectrum in which fairness assumptions are preserved. The
identified equivalences allow for (compositional) reasoning about correctness
incorporating fairness assumptions.
