On the omega values of generators of embedding dimension-three numerical monoids generated by an interval

Abstract

We offer a formula to compute the omega values of the generators of the numerical monoid S = k, k + 1, k + 2 where k is a positive integer greater than 2.

Measuring primality in numerical semigroups with embedding dimension three

“…In this paper we compute the ω-values of the generators in the embedding dimension three case (Theorems 4.6 and 4.9). As a by-product, in Theorem 6.2 we confirm the conjecture cited above in [8]. Additionally, we relate, for embedding dimension three, the ω-value of a numerical semigroup to its catenary degree (cf.…”

“…The arithmetic of non-unique factorizations in rings and monoids has been a popular topic in the recent mathematical literature. We focus on extending the results in [3,4,8], where the ω-function, an arithmetic measure of how far an element is from being prime (cf. Sec.…”

“…When S is a numerical semigroup with minimal set of generators {n 1 , n 2 , n 3 }, with n 1 < n 2 < n 3 , there are 13 possible inequalities involving the ω(n i ) (for instance ω(n 1 ) ≤ ω(n 2 ) < ω(n 3 )). Examples of eight of these orderings were given in [3], and two more were given in [8]. The authors in [8] conjectured that the last three orderings are not possible.…”

“…Examples of eight of these orderings were given in [3], and two more were given in [8]. The authors in [8] conjectured that the last three orderings are not possible. In this paper we compute the ω-values of the generators in the embedding dimension three case (Theorems 4.6 and 4.9).…”

On the linearity of ω-primality in numerical monoids

“…, n k . For n ∈ Γ, define ω(n) = m if m is the smallest positive integer with the property that whenever a ∈ N k satisfies k i=1 a i n i − n ∈ Γ with | a| > m, there exist a b ∈ N k with b i ≤ a i for each i such that k i=1 b i n i − n ∈ Γ and | b| ≤ m. The notion of a bullet was first introduced in [7]. Throughout this paper, we use bullets extensively to study the ω-function of numerical monoids.…”