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On some properties for cofiniteness of submonoids and ideals of an affine semigroup

Carmelo Cisto

Abstract

Let $S$ and $\mathcal{C}$ be affine semigroups in $\mathbb{N}^d$ such that $S\subseteq \mathcal{C}$. We provide a characterization for the set $\mathcal{C}\setminus S$ to be finite, together with a procedure and computational tools to check whether such a set is finite and, if so, compute its elements. As a consequence of this result, we provide a characterization for an ideal $I$ of an affine semigroup $S$ so that $S\setminus I$ is a finite set. If so, we provide some procedures to compute the set $S\setminus I$.

On some properties for cofiniteness of submonoids and ideals of an affine semigroup

Abstract

Let and be affine semigroups in such that . We provide a characterization for the set to be finite, together with a procedure and computational tools to check whether such a set is finite and, if so, compute its elements. As a consequence of this result, we provide a characterization for an ideal of an affine semigroup so that is a finite set. If so, we provide some procedures to compute the set .
Paper Structure (4 sections, 11 theorems, 14 equations)

This paper contains 4 sections, 11 theorems, 14 equations.

Table of Contents

  1. Cofinite submonoids of an affine semigroup
  2. Cofinite ideals of an affine semigroup
  3. An approach using commutative algebra
  4. A remark on Apéry sets

Key Result

Proposition 1

Let $\mathcal{C}\subseteq \mathbb{N}^d$ be an affine semigroup and $S\subseteq \mathcal{C}$ be a $\mathcal{C}$-cofinite submnoid. Then $S$ is finitely generated.

Theorems & Definitions (24)

  • Definition 1
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3: Analele
  • Theorem 4
  • proof
  • Example 5
  • Proposition 6
  • ...and 14 more