On nearly Gorenstein affine semigroups

We describe the canonical module of a simplicial affine semigroup ring $\mathbb{K}[S]$ and its trace ideal. As a consequence, we characterize when $\mathbb{K}[S]$ is nearly Gorenstein in terms of arithmetic properties of the semigroup $S$. Then, we find some bounds for the Cohen-Macaulay type of $\mathbb{K}[S]$ when it is nearly Gorenstein. In particular, if it has codimension at most three, we prove that the Cohen-Macaulay type is at most three and this bound is sharp.