Boundedness of type for almost symmetric generalized numerical semigroups at fixed embedding dimension

Determine whether there exists a universal upper bound on the type t(K[S]) for the class of almost symmetric generalized numerical semigroups S ⊆ N^d when the embedding dimension e is fixed; equivalently, ascertain whether there exists a function B(e) such that for every almost symmetric generalized numerical semigroup S of embedding dimension e, one has t(S) ≤ B(e).

Background

The paper studies extremal behavior of invariants of C-semigroups, including type and reduced type, and proves that for generalized numerical semigroups with fixed embedding dimension, type can be unbounded in general. However, the boundedness of type is sensitive to structural subclasses.

In the numerical semigroup literature, the boundedness of type within the subclass of almost symmetric semigroups is a known open problem, and for low embedding dimensions (e ≤ 5) the type is bounded (cited results by Moscariello). The authors extend this open problem to generalized numerical semigroups (affine semigroups with full cone C = N^d).

Almost symmetric generalized numerical semigroups are defined via pseudo-Frobenius elements and a term-order characterization that the paper shows to be independent of the choice of term order. In this subclass, whether type admits an upper bound solely in terms of the fixed embedding dimension remains unresolved.