The realization problem for delta sets of numerical semigroups
S Colton, N Kaplan - 2017 - projecteuclid.org
The delta set of a numerical semigroup S, denoted Δ(S), is a factorization invariant that
measures the complexity of the sets of lengths of elements in~ S. We study the following
problem: Which finite sets occur as the delta set of a numerical semigroup S? It is known that
\minΔ(S)=\gcdΔ(S) is a necessary condition. For any two-element set {d,td\} we produce a
semigroup~ S with this delta set. We then show that, for t≥2, the set {d,td\} occurs as the
delta set of some numerical semigroup of embedding dimension~ 3 if and only if t=2.
measures the complexity of the sets of lengths of elements in~ S. We study the following
problem: Which finite sets occur as the delta set of a numerical semigroup S? It is known that
\minΔ(S)=\gcdΔ(S) is a necessary condition. For any two-element set {d,td\} we produce a
semigroup~ S with this delta set. We then show that, for t≥2, the set {d,td\} occurs as the
delta set of some numerical semigroup of embedding dimension~ 3 if and only if t=2.
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