Polytopal affine semigroups with holes deep inside

TL;DR

This paper constructs a specific lattice 3-simplex with an associated affine semigroup that is not normal, yet all holes are deeply embedded inside, far from the facets.

Contribution

It introduces a method to construct polytopes with deep holes in their affine semigroups, advancing understanding of non-normal semigroups.

Findings

Constructed a lattice 3-simplex with deep holes in its affine semigroup.

Showed the affine semigroup is not normal despite deep holes.

Demonstrated the existence of non-normal semigroups with holes far from facets.

Abstract

Given a non-negative integer k, we construct a lattice 3-simplex P with the following property: The affine semigroup Q_P associated to P is not normal, and every element $q \in \sat{Q}_P \setminus Q_P$ has lattice distance at least k above every facet of Q_P.