Several recent papers have examined a rational polyhedron P m whose integer points are in bijection with the numerical semigroups (cofinite, additively closed substets of the non-negative integers) containing m. A combinatorial description of the faces of P m was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of P m containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of P m . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of P m .
A numerical semigroup S is an additive subsemigroup of the nonnegative integers, containing zero, with finite complement. Its multiplicity m is its smallest nonzero element. The Apéry set of S is the set of elements Ap(S) = {n ∈ S : n − m ∈ S}. Fixing a numerical semigroup, we ask how many elements of its Apéry set have nonunique factorization and define several new invariants.
For affine monoids of dimension 2 with embedding dimension 2 and 3, we study the problem of determining when a vector is an element of the monoid, and the problem of determining the elasticity of a monoid element.
We call a (not necessarily planar) embedding of a graph G in the plane sequential if its vertices lie in Z 2 and the line segments between adjacent vertices contain no interior integer points. In this note, we prove (i) a graph G has a sequential embedding if and only if G is 4-colorable, and (ii) if G is planar, then G has a sequential planar embedding.
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