On the additive structure of algebraic valuations of polynomial semirings

In this paper, we study factorizations in the additive monoids of positive algebraic valuations N0[α] semiring polynomials N0[X] using a methodology introduced by D. Anderson, F. and M. Zafrullah 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin determining when atomic, give an explicit description its set An finite factorization (FFM) has only finitely many (up to order associates), it bounded (BFM) for there bound number irreducibles (counting repetitions) each factorizations. show that, N0[α], property being BFM FFM are equivalent ascending chain condition on principal ideals (ACCP). Finally, various characterizations be unique (UFM), two them terms minimal polynomial α. The properties generated, half-factorial, length-factorial also investigated along way.