  We introduce the notion of hereditary G-compactness (with respect to
interpretation). We provide a sufficient condition for a poset to not be
hereditarily G-compact, which we use to show that any linear order is not
hereditarily G-compact. Assuming that a long-standing conjecture about unstable
NIP theories holds, this implies that an NIP theory is hereditarily G-compact
if and only if it is stable (and by a result of Simon, this holds
unconditionally for $\aleph_0$-categorical theories). We show that if $G$ is
definable over $A$ in a hereditarily G-compact theory, then
$G^{00}_A=G^{000}_A$.
  We also include a brief survey of sufficient conditions for G-compactness,
with particular focus on those which can be used to prove or disprove
hereditary G-compactness for some (classes of) theories.

Logic