  We present Monte Carlo simulations on a new class of lattice models in which
the degrees of freedom are elements of an abelian or non-abelian finite
symmetry group G, placed on directed edges of a two-dimensional lattice. The
plaquette group product is constrained to be the group identity. In contrast to
discrete gauge models (but similar to past work on height models) only elements
of symmetry-related subsets S of G are allowed on edges. These models have
topological sectors labeled by group products along topologically non-trivial
loops. Measurement of relative sector probabilities and the distribution of
distance between defect pairs are done to characterize the types of order
(topological or quasi-LRO) exhibited by these models. We present particular
models in which fully local non-abelian constraints lead to global topological
liquid properties.
