  A spin-1/2 two-leg ladder with four-spin ring exchange is studied by
quantized Berry phases, used as local order parameters. Reflecting local
objects, non-trivial ($\pi$) Berry phase is founded on a rung for the
rung-singlet phase and on a plaquette for the vector-chiral phase. Since the
quantized Berry phase is topological invariant for gapped systems with the time
reversal symmetry, topologically identical models can be obtained by the
adiabatic modification. The rung-singlet phase is adiabatically connected to a
decoupled rung-singlet model and the vector-chiral phase is connected to a
decoupled vector-chiral model. Decoupled models reveals that the local objects
are a local singlet and a plaquette singlet respectively.
