Let be a class of processes. I assume that for each process
a set of (partial) runs is defined, equipped
with a prefix ordering , and that with each run (or *
execution*) is associated its observable content .

A first idea of what could be the branching structure of a process , is the pomset with representative . This I
will call the * strict branching structure* of **p**, as it
corresponds with the idea that the branching of all, possibly
indistinguishable, runs is taken into account. In order to
arrive at a less strict definition, one needs to identify certain
runs.

This is the intermediate variant of three possible generalizations of
the concept of a congruence from algebras to algebraic structures with
relations. A * strong* congruence would even require that , so
that can simply be defined by . A * weak* congruence
on the other hand doesn't require anything for , and
needs to be defined by .

Obviously a coarsest congruence on a pomset exists, as the union of arbitrary many congruences is again a congruence. The following observations are meant as tool for deciding whether or not a model or equivalence respects branching time.

** Proof: **
This follows immediately from the following two facts, whose proofs are
straightforward.

- If is the coarsest and any congruence on an elpo , and is the coarsest congruence on , then .
- If and is a congruence on , then there is a congruence on such that . Moreover, is the coarsest iff is.