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We characterize the submonoids $M$ of the additive monoid $\Q_+$ ofnonnegative rational numbers for which the irreducible and the prime elementsin the monoid domain $F[X;M]$ coincide. We present a diagram of implicationsbetween some types of submonoids of $\Q_+$, with a precise position of themonoids $M$ with this property.

For which additive submonoids $M$ of nonnegative rationals is $F[X;M]$ AP?