On the set of atoms and strong atoms in additive monoids of cyclic semidomains

Let M be a cancellative and commutative monoid. A non-invertible element of M is called an atom (or irreducible element) if it cannot be factored into two non-invertible elements, while an atom a of M is called strong if an has a unique factorization in M for every n∈N. The monoid M is atomic if every non-invertible element factors into finitely many atoms (repetitions allowed). For an algebraic number α, we let Mα denote the additive monoid of the subsemiring N0[α] of C. The atomic structure of Mα reflects intricate interactions between algebraic number theory and additive semigroup theory. For m,n∈N0∪{∞} (with m≤n), the pair (m,n) is called realizable if there exists an algebraic number α∈C such that Mα has m strong atoms and n atoms. Our primary goal is to identify classes of realizable pairs with the long-term goal of providing a complete description of the full set of realizable pairs.