  Atom losses occur naturally during cold atoms experiments. Since this
phenomenon is unavoidable, it is important to understand its effect on the
remaining atoms. Here we study a gas of hard-core bosons on a lattice subject
to $K$-body losses (where $K=1,2,3,\dots$ is the number of atoms lost in each
loss event), and in particular we investigate the effect of losses on the
rapidity distribution $\rho(k)$ of the atoms. Under the assumption that losses
are weak enough so that the system relaxes between two loss events, we are able
to determine the loss functional $F[\rho](k)$ encoding the loss process for
$K$-body losses. We derive closed expressions for the cases of one- and
two-body losses, and show their effects on the evolution of the total number of
particles. Then we add a harmonic trapping potential and study the evolution of
the position-dependent rapidity distribution of this system by solving
numerically the evolution equation for one-, two- and three-body losses.
