  Monte Carlo simulations of mono-- and polydisperse two--dimensional crystals
are reported. The particles in the studied system, interacting through
hard--core repulsive Yukawa potential, form a solid phase of hexagonal lattice.
The elastic properties of crystalline Yukawa systems are determined in the
$NpT$ ensemble with variable shape of the periodic box. Effects of the Debye
screening length ($\kappa^{-1}$), contact value of the potential ($\epsilon$),
and the size polydispersity of particles on elastic properties of the system
are studied. The simulations show that the polydispersity of particles strongly
influences the elastic properties of the studied system, especially on the
shear modulus. It is also found that the elastic moduli increase with density
and their growth rate depends on the screening length. Shorter screening length
leads to faster increase of elastic moduli with density and decrease of the
Poisson's ratio. In contrast to its three-dimensional version, the studied
system is non-auxetic, i.e. shows positive Poisson's ratio.
