Let M be a state machine 
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There are two reasonable ways to define what the behavior of a state machine is. One way (``event-based'' or ``action-based'') says what is observable are a machine's actions; the other (``state-based'') says what is observable are a machine's states. Which way you prefer is philosophical. Here are two alternative definitions of what a trace is:
Finally, we can define what the behavior of a machine is.
Behaviors are prefix-closed, which means, for a given behavior, B: (1) The empty trace is in Beh(M); and (2) if a trace is in B then any prefix of that trace is in B.
In other work on state machine models, behaviors do not have or are not assumed to have the prefix-closure property.