  Sound behavioral equations on open terms may become unsound after
conservative extensions of the underlying operational semantics. Providing
criteria under which such equations are preserved is extremely useful; in
particular, it can avoid the need to repeat proofs when extending the specified
language.
  This paper investigates preservation of sound equations for several notions
of bisimilarity on open terms: closed-instance (ci-)bisimilarity and
formal-hypothesis (fh-)bisimilarity, both due to Robert de Simone, and
hypothesis-preserving (hp-)bisimilarity, due to Arend Rensink. For both
fh-bisimilarity and hp-bisimilarity, we prove that arbitrary sound equations on
open terms are preserved by all disjoint extensions which do not add labels. We
also define slight variations of fh- and hp-bisimilarity such that all sound
equations are preserved by arbitrary disjoint extensions. Finally, we give two
sets of syntactic criteria (on equations, resp. operational extensions) and
prove each of them to be sufficient for preserving ci-bisimilarity.
