arXiv subjects

R. Tapia-Ramos

Publications and source records attributed to R. Tapia-Ramos.

7 recordsLinked to original sources

arXiv2026

A note on strong affine semigroups

This work introduces and studies strong affine semigroups, extending the notion of strong numerical semigroups to the higher-dimensional setting. We show that non-numerical strong affine semigroups present structural differences with respect to strong numerical semigroups. Special attention is devoted to strong $\mathcal C$-semigroups. We prove that the family of strong $\mathcal C$-semigroups with a given set of multiplicities $E$ admits a maximal element and has a tree structure. We characterize when this family is finite and provide an algorithm to compute all such semigroups up to a fixed genus. We also introduce the notion of special strong affine semigroups and obtain refined versions of several previous results. Finally, we study toric ideals arising from strong affine semigroups, determining their indispensable monomials and Betti elements for several families.

math.AC↗

arXiv2025

On some affine semigroups characterized by a finite-state automata

This work introduces a new kind of affine semigroups called $P$-semigroups. Within the framework of $\mathcal C$-semigroups, we define a finite-state automaton associated to them. Moreover, this automaton determines whether a $\mathcal C$-semigroup is a $P$-semigroup, which represents a bridge between affine semigroups and Discrete Mathematics. Furthermore, some algorithms for computing all the $P$-semigroups with a fixed Frobenius element, genus, or multiplicity are provided.

math.AC↗

arXiv2025

Affine semigroups without consecutive small elements

An $\mathcal{A}$-semigroup is a numerical semigroup without consecutive small elements. This work generalizes this concept to finite-complement submonoids of an affine cone $\mathcal{C}$. We develop algorithmic procedures to compute all $\mathcal{A}$-semigroups with a given Frobenius element (denoted by $\mathcal{A}(f)$), and with fixed Frobenius element and multiplicity. Moreover, we analyze the $\mathcal{A}(f)$-systems of generators. Furthermore, we study $\mathcal{A}$-numerical semigroups with maximal embedding dimension, fixed Frobenius number and multiplicity, providing an algorithm for their computation and a graphical classification.

math.AC↗

arXiv2024

A computational approach to the study of finite-complement submonids of an affine cone

Let $\mathcal{C}\subseteq \mathbb{N}^p$ be an integer cone. A $\mathcal{C}$-semigroup $S\subseteq \mathcal{C}$ is an affine semigroup such that the set $\mathcal{C}\setminus S$ is finite. Such $\mathcal{C}$-semigroups are central to our study. We develop new algorithms for computing $\mathcal{C}$-semigroups with specified invariants, including genus, Frobenius element, and their combinations, among other invariants. To achieve this, we introduce a new class of $\mathcal{C}$-semigroups, termed $\mathcal{B}$-semigroups. By fixing the degree lexicographic order, we also research the embedding dimension for both ordinary and mult-embedded $\mathbb{N}^2$-semigroups. These results are applied to test some generalizations of Wilf's conjecture.

math.AC↗

arXiv2024

C-semigroups with its induced order

Let $C\subset\mathbb{N}^p$ be an integer polyhedral cone. An affine semigroup $S\subset C$ is a $ C$-semigroup if $| C\setminus S|<+\infty$. This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to $C$-semigroups.

math.AC↗

arXiv2024

On ideals of affine semigroups and affine semigroups with maximal embedding dimension

Let $S\subseteq \mathbb N^p$ be a semigroup, any $P\subseteq S$ is an ideal of $S$ if $P+S\subseteq P$, and an $I(S)$-semigroup is the affine semigroup $P\cup \{0\}$, with $P$ an ideal of $S$. We characterise the $I(S)$-semigroups and the ones that also are $\mathcal C$-semigroups. Moreover, some algorithms are provided to compute all the $I(S)$-semigroups satisfying some properties. From a family of ideals of $S$, we introduce the affine semigroups with maximal embedding dimension, characterising them and describing some families.

math.AC↗

arXiv2024

On the quotient of affine semigroups by a positive integer

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.

math.AC↗