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A general approach for defining behavioral preorders over process terms as the maximal precongruences induced by basic observables is examined. Three different observables that provide information about the initial communication capabilities of processes and about the possibility that processes get engaged in divergent computations will be considered. We show that the precongruences induced by our basic observables coincide with intuitive and/or widely studied behavioral preorders. In particular, we retrieve in our setting themust preorderof De Nicola and Hennessy and thefair/should preorderintroduced by Cleaveland and Natarajan and by Brinksma, Rensink, and Vogler. A new form of testing preorder, which we callsafe-must, also emerges. The alternative characterizations we offer shed light on the differences between these pre- orders and on the role played in their definition by tests for divergence.