  Motivated by recent experiments, we compare the Tonks (i.e. hard-core boson
gas) regime achieved in an optical lattice with the Tonks regime of a
one-dimensional Bose gas in the continuum. For the lattice gas, we compute the
local (i.e. on-site) two-body correlations as a function of temperature and the
filling of the lattice. It is found that this function saturates to a constant
value with increasing temperature. Furthermore, the parameter that
characterizes the long-distance correlations in the lattice Tonks regime is
also obtained, showing that on the lattice the long-distance correlations enter
the Tonks regime more rapidly than in the continuum.
