Classical height models with topological order

I discuss a family of statistical-mechanics models in which (some classes of) elements of a finite group G occupy the (directed) edges of a lattice; the product around any plaquette is constrained to be the group identity e. Such a model may possess topological order, i.e. its equilibrium ensemble has distinct, symmetry-related thermodynamic components that cannot be distinguished by any local order parameter. In particular, if G is a non-Abelian group, the topological order may be non-Abelian. Criteria are given for the viability of particular models, in particular for Monte Carlo updates.