We consider various two-leg ladder models exhibiting gapped phases. All of
these phases have short-ranged valence bond ground states, and they all exh
ibit string order. However, we show that short-ranged valence bond ground s
tates divide into two topologically distinct classes, and as a consequence,
there exist two topologically distinct types of string order. Therefore, n
ot all gapped phases belong to the same universality class. We show that ph
ase transitions occur when we interpolate between models belonging to diffe
rent topological classes, and we study the nature of these transitions.