Semigroups that are Factorial from Inside or from Outside

Abstract

The multiplicative semigroup of natural numbers ℕ = {1, 2,...} is factorial, i.e. every number ≠ 1 has a unique factorization into irreducible numbers (up to the ordering of the factors). Subsemigroups of ℕ however need not be factorial. Consider for example the well-known Hilbert semigroup H = 4ℤ₊ + 1 = {4n + 1 ∣ n∈ℕ₊} with the multiplication operation. (Thereby ℕ₊ = {n∈ℕ | n ≥ 0}, ℕ = {0, ±1, ±2,...}.) The number 441, e.g, factorizes in H in the two essentially different ways 21 ·21 = 441 = 9 ·49 into numbers which are irreducible within H. (Obviously none of these irreducible numbers can be prime within H.)