  It has been recently shown that the presence of topological frustration,
induced by periodic boundary conditions in an antiferromagnetic $XY$ chain made
of an odd number of spins, prevents the realization of a perfectly staggered
local order. Starting from this result and exploiting a recently introduced
approach which enables the direct calculation of the expectation value of any
operator with support over a finite range of lattice sites, in this work we
investigate the possible fates of local orders. We show that, regardless of the
variety of possible situations, they can be all arranged in two different
cases. A system admits a finite local order only if the ground state is
degenerate, with at least two elements whose momenta differ, in the
thermodynamic limit, by $\pi$, and this order breaks translational symmetry. In
all other cases, any local order decays to zero, algebraically (or faster) in
the chain length. Moreover, we show that, in some cases, which of the two
possibilities is realized, may depend on the sequence of chain lengths with
which the thermodynamic limit is reached. These results are established both
analytically and by exact diagonalization and illustrated through examples.
