  We study the halo mass function in the presence of the kurtosis type of
primordial non-Gaussianity. The kurtosis corresponds to the trispectrum as
defined in Fourier space. The primordial trispectrum is commonly characterized
by two parameters, $\tau_{\rm NL}$ and $g_{\rm NL}$. As applications of the
derived non-Gaussian mass function, we consider the effect on the abundance of
void structure, the effect on early star formation and on formation of the most
massive object at high redshift. We show that by comparing the effects of
primordial non-Gaussianity on cluster abundance with that on void abundance, we
can distinguish between the skewness and the kurtosis types of primordial
non-Gaussianity. As for early star formation, we show that the kurtosis type of
primordial non-Gaussianity seems not to affect the reionization history of the
Universe on average. However, at high redshifts (up to $z\simeq 20$) such
non-Gaussianity does somewhat affect the early stages of reionization.
