A global checkpoint is a set of local checkpoints, one per process. The traditional consistency criterion for global checkpoints states that a global checkpoint is consistent if it does not include messages received and not sent. The paper investigates other consistency criteria, transitlessness, and strong consistency. A global checkpoint is transitless if it does not exhibit messages sent and not received. Transitlessness can be seen as a dual of traditional consistency. Strong consistency is the addition of transitlessness to traditional consistency. The main result of the paper is a statement of the necessary and sufficient condition answering the following question: given an arbitrary set of local checkpoints, can this set be extended to a global checkpoint that satisfies (where P is traditional consistency, transitlessness, or strong consistency). From a practical point of view, this condition, when applied to transitlessness, is particularly interesting as it helps characterize which messages do not need to be recorded by checkpointing protocols.