Determinantal defining ideals of numerical semigroup rings with structured pseudo-Frobenius sets

Numerical semigroup rings provide a natural meeting point between additive semigroup theory and commutative algebra, while pseudo-Frobenius numbers encode key information about the type, symmetry, and defining relations. This paper investigates when the defining ideal $ I_{H} $ of a numerical semigroup ring $ K[H] $ can be described by the maximal minors of a structured matrix determined by the combinatorics of $ \operatorname{PF}(H) $. For a numerical semigroup $ H = \langle n_1, \, \ldots, \, n_e \rangle $ the defining ideal always has a height $ e-1 $; we therefore study $ I_{H} $ through a candidate $ 2\times e $ matrix, whose ideal of maximal ($ 2\times 2 $) minors has the correct expected height $ e-1 $, and we test the equality $ I_{H} = I_{2}(M) $ by degree arguments, an explicit height comparison, and the Eagon-Northcott resolution. We prove that the numerical semigroups $ H = \langle n, \, n+1, \, n+2, \, n+3 \rangle $ with $ n\equiv 1\pmod 3 $ form an infinite family with the consecutive (arithmetic) pseudo-Frobenius set $ \operatorname{PF}(H) = \left\{ \operatorname{F}(H)-2, \, \operatorname{F}(H)-1, \, \operatorname{F}(H) \right\} $ and a determinantal defining ideal; for $ H = \langle 4, \, 5, \, 6, \, 7 \rangle $, the defining ideal equals the ideal of $ 2\times 2 $ minors of an explicit $ 2\times 4 $ matrix, giving Betti numbers $ \bigl(\beta_{0}, \, \beta_{1}, \, \beta_{2}\bigr) = (6, \, 8, \, 3) $. More generally the Eagon-Northcott complex of a $ 2\times e $ matrix yields

$ \beta_{i}(I_{H}) = (i+1)\binom{e}{i+2}, $

so that the number of defining binomials is $ \binom{e}{2} $. We also give a simple necessary numerical obstruction in embedding dimensions $ e\equiv1\pmod 3 $: If the defining ideal is the ideal of maximal minors of an $ H $-homogeneous $ 2\times e $ matrix, then the sum of the $ H $-degrees of its minimal binomial generators is divisible by $ 3 $. In particular, this gives an effective obstruction in embedding dimension four. Using this obstruction, we exhibit numerical semigroups whose defining ideals share the Eagon-Northcott Betti numbers yet are provably not determinantal. All examples are computed and verified with Singular; the code and its output are reported in full. The results clarify how the internal structure of $ \operatorname{PF}(H) $ governs, and sometimes fails to govern, the determinantal presentation of $ K[H] $.