We study the semiring \(\mathbb {N}_0[\alpha ]\) as an additive monoid where \(\alpha \) is a positive real algebraic number. In the atomic case, the atoms of \(\mathbb {N}_0[\alpha ]\) are precisely the powers \(\alpha ^n\) up to a certain nonnegative integer n, and finite generation is governed by divisibility of the minimal polynomial by a negative-tail polynomial. Our first main result gives a complete characterization when the minimal polynomial has the form \(\mathfrak {m}_\alpha (X)=p_\alpha (X)-c\) with \(c\in \mathbb {N}\) . Our second main result shows that finite generation forces \(\alpha \) to be a weak Perron number, and proves a converse under the additional assumptions that \(\alpha \) is an algebraic integer and the unique positive conjugate of its minimal polynomial. As an application, we analyze cubic minimal polynomials and obtain a partial classification of rank-3 monoids \(\mathbb {N}_0[\alpha ]\) by generation and factorization type, including coefficient constraints, non-length-factoriality results for a large family, and examples with prescribed numbers of atoms.
