  A two-dimensional lattice gas model is proposed. The ground state of this
model with a fixed density is neither periodic nor quasi-periodic. It also
depends on system size in an irregular manner. On the other hand, it is ordered
in the sense that the entropy density is zero in the thermodynamic limit. The
existence of a thermodynamic transition associated with such irregularly
ordered ground states is conjectured from a duality relation for a
thermodynamic function. This conjecture is supported by a phenomenological
argument and numerical experiments.
