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On the quotient of affine semigroups by a positive integer

J. I. García-García, R. Tapia-Ramos, A. Vigneron-Tenorio

Abstract

This work delves into the {\it quotient of an affine semigroup by a positive integer}, exploring its intricate properties and broader implications. We unveil an {\it associated tree} that serves as a valuable tool for further analysis. Moreover, we successfully generalize several key irreducibility results, extending their applicability to the more general class of $\mathcal C$-semigroup quotients. To shed light on these concepts, we introduce the novel notion of an {\it arithmetic variety of affine semigroups}, accompanied by illuminating examples that showcase its power.

On the quotient of affine semigroups by a positive integer

Abstract

This work delves into the {\it quotient of an affine semigroup by a positive integer}, exploring its intricate properties and broader implications. We unveil an {\it associated tree} that serves as a valuable tool for further analysis. Moreover, we successfully generalize several key irreducibility results, extending their applicability to the more general class of -semigroup quotients. To shed light on these concepts, we introduce the novel notion of an {\it arithmetic variety of affine semigroups}, accompanied by illuminating examples that showcase its power.
Paper Structure (5 sections, 23 theorems, 12 equations, 5 figures, 1 algorithm)

This paper contains 5 sections, 23 theorems, 12 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Preliminars
  2. Some families of affine semigroups closed under the quotient by a positive integer
  3. Computing $\mathcal{D}_d(S)$ and building a tree on an arithmetic variety
  4. Some results about irreducibility over the quotients of $\mathcal{C}$-semigroups
  5. Arithmetic varieties of affine semigroups

Key Result

Proposition 1

AffineSmgp Let $S$ be a $\mathcal{C}$-semigroup and $x\in \mathcal{C}$. Then, $x\notin S$ if and only if there exists $f\in PF(S)$ such that $f-x\in S.$

Figures (5)

  • Figure 1: $\mathcal{C}$-semigroup minimally generated by \ref{['semigrupo_ejemplo1']}.
  • Figure 2: $T(f,\emptyset)$.
  • Figure 3: Some elements in $\mathcal{D}_3(S)$.
  • Figure 4: $G_{\mathcal{A}_{(4,2)},2}$.
  • Figure 5: A symmetric semigroup $T$ with $S=\frac{T}{2}$ obtained from Theorem \ref{['thrmQuoSim']}.

Theorems & Definitions (43)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 33 more