Searcharxiv⌕ Search

arXiv subjects

Carmelo Cisto

Publications and source records attributed to Carmelo Cisto.

At least 19 recordsLinked to original sources
arXiv2025

Positioned and primary positioned $\mathcal{C}$-semigroups

Let $\mathcal{C}$ be a positive integer cone and $k\in \mathcal{C}$. A $\mathcal{C}$-semigroup $S$ is $k$-positioned if for every $h\in \mathcal{C}\setminus S$ we have that $k-h$ belongs to $S$. In this work, we focus on this family of semigroups and introduce primary positioned $\mathcal{C}$-semigroups, characterizing a subfamily of them through the perspective of irreducibility. Furthermore, we provide some procedures to compute all such semigroups, describing a family of graphs containing all the primary positioned $\mathcal{C}$-semigroups for a fixed $k\in \mathcal{C}$.

math.CO↗
arXiv2025

Buchsbaumness of finite complement simplicial affine semigroups

In this article, we classify all Buchsbaum simplicial affine semigroups whose complement in their (integer) rational polyhedral cone is finite. We show that such a semigroup is Buchsbaum if and only if its set of gaps is equal to its set of pseudo-Frobenius elements. Furthermore, we provide a complete structure of these affine semigroups. In the case of affine semigroups with maximal embedding dimension, we provide an explicit formula for the cardinality of the minimal presentation in terms of the number of extremal rays, the embedding dimension, and the genus. Finally, we observe that, unlike the complete intersection, Cohen-Macaulay, and Gorenstein properties, the Buchsbaum property is not preserved under gluing of affine semigroups.

math.AC↗
arXiv2025

Symmetric generalized numerical semigroups in $\mathbb{N}^d$ with embedding dimension $2d+1$

In this article, we classify all symmetric generalized numerical semigroups in $\mathbb{N}^d$ of embedding dimension $2d+1$. Consequently, we show that in this case the property of being symmetric is equivalent to have a unique maximal gap with respect to natural partial order in $\mathbb{N}^d$. Moreover, we deduce that when $d>1$, there does not exist any generalized numerical semigroup of embedding dimension $2d+1$, which is almost symmetric but not symmetric.

math.AC↗
arXiv2025

On Cohen-Macaulay non-prime collections of cells

In this paper we investigate Cohen-Macaulayness, Gorensteinness and the Hilbert-Poincaré series for some classes of non-prime collections of cells. In particular, we show that all closed path polyominoes are Cohen-Macaulay and we characterize those that are Gorenstein.

math.AC↗
arXiv2025

On unboundedness of some invariants of $\mathcal{C}$-semigroups

In this article, we first prove that the type of an affine semigroup ring is equal to the number of maximal elements of the Apéry set with respect to the set of exponents of the monomials, which form a maximal regular sequence. Further, we consider $\mathcal{C}$-semigroups in $\mathbb{N}^d$ and prove that the notions of symmetric and almost symmetric $\mathcal{C}$-semigroups are independent of term orders. We further investigate the conductor and the Apéry set of a $\mathcal{C}$-semigroup with respect to a minimal extremal ray. Building upon this, we extend the notion of reduced type to $\mathcal{C}$-semigroups and study its extremal behavior. For all $d$ and fixed $e \geq 2d$, we give a class of $\mathcal{C}$-semigroups of embedding dimension $e$ such that both the type and the reduced type do not have any upper bound in terms of the embedding dimension. We further explore irreducible decompositions of a $\mathcal{C}$-semigroup and give a lower bound on the irreducible components in an irreducible decomposition. Consequently, we deduce that for each positive integer $k$, there exists a $\mathcal{C}$-semigroup $S$ such that the number of irreducible components of $S$ is at least $k$.

math.AC↗
arXiv2024

Generalized Numerical semigroups up to isomorphism

A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ with finite complement in it. We characterize isomorphisms between these monoids in terms of permutation of coordinates. Considering the equivalence relation that identifies the monoids obtained by the action of a permutation and establishing a criterion to select a representative from each equivalence class, we define some procedures for generating the set of all generalized numerical semigroups of given genus up to isomorphism. Finally, we present computational data and explore properties related to the number of generalized numerical semigroups of a given genus up to isomorphism.

math.CO↗
arXiv2024

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

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 $S\setminus I$.

math.AC↗
arXiv2023

Polyocollection ideals and primary decomposition of polyomino ideals

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can be used to compute a primary decomposition of non-prime polyomino ideals. Furthermore, we give a description of the minimal primary decomposition of non-prime closed path polyominoes. In particular, for such a class of polyominoes, we characterize the set of all zig-zag walks and show that the minimal prime ideals have a very nice combinatorial description.

math.AC↗
arXiv2023

Hilbert-Poincaré series and Gorenstein property for some non-simple polyominoes

Let $\mathcal{P}$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper we give a combinatorial interpretation of the $h$-polynomial of $K[\mathcal{P}]$, showing that it is the rook polynomial of $\mathcal{P}$. It is known by Rinaldo and Romeo (2021), that if $\mathcal{P}$ is a simple thin polyomino then the $h$-polynomial is equal to the rook polynomial of $\mathcal{P}$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert-Poincaré series of the coordinate ring attached to a closed path $\mathcal{P}$ having no zig-zag walks, as a combination of the Hilbert-Poincaré series of convenient simple thin polyominoes. As a consequence we prove that the Krull dimension is equal to $\vert V(\mathcal{P})\vert -\mathrm{rank}\, \mathcal{P}$ and the regularity of $K[\mathcal{P}]$ is the rook number of $\mathcal{P}$. Finally we characterize the Gorenstein prime closed paths, proving that $K[\mathcal{P}]$ is Gorenstein if and only if $\mathcal{P}$ consists of maximal blocks of length three.

math.AC↗
arXiv2022

On some classes of generalized numerical semigroups

A generalized numerical semigroup is a submonoid of $\mathbb{N}^d$ with finite complement in it. In this work we study some properties of three different classes of generalized numerical semigroups. In particular, we prove that the first class satisfies a generalization of Wilf's conjecture, by introducing a generalization of a well-known sufficient condition for Wilf's conjecture in numerical semigroups, that involves the type of the semigroup. Partial results for Wilf's generalized conjecture are obtained also for the other two classes, and some open questions are provided.

math.CO↗
arXiv2022

On some numerical semigroup transforms

In this paper we introduce a particular semigroup transform $\mathcal{A}$ that fixes the invariants involved in Wilf's conjecture, except the embedding dimension. It also allows one to arrange the set of not ordinary and not irreducible numerical semigroups in a family of rooted trees. We study also another transform, having similar features, that has been introduced by Bras-Amorós, and we make a comparison of them. In particular we study the behaviour of the embedding dimension under the action of such transforms, providing some consequences concerning Wilf's conjecture.

math.CO↗
arXiv2022

Primality of weakly connected collections of cells and weakly closed path polyominoes

In this paper we study the primality of weakly connected collections of cells, showing that the ideal generated by inner 2-minors attached to a weakly connected and simple collection of cells is the toric ideal of the edge ring of a weakly chordal bipartite graph. As an application of this result we characterize the primality of the polyomino ideals of weakly closed paths, a new class of non simple polyominoes.

math.AC↗
arXiv2022

On Gröbner bases and Cohen-Macaulay property of closed path polyominoes

In this paper we introduce some monomial orders for the class of closed path polyominoes and we prove that the set of the generators of the polyomino ideal attached to a closed path forms the reduced Gröbner basis with respect to these monomial orders. It is known that the polyomino ideal attached to a closed path containing an L-configuration or a ladder of at least three steps, equivalently having no zig-zag walks, is prime. As a consequence, we obtain that the coordinate ring of a closed path having no zig-zag walks is a normal Cohen-Macaulay domain.

math.AC↗
arXiv2021

Primality of closed path polyominoes

In this paper we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths.

math.CO↗
arXiv2020

On almost-symmetry in generalized numerical semigroups

In this work we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that this class yields a new family of Frobenius generalized numerical semigroups and extends the class of irreducible generalized numerical semigroups. This investigation allows us to provide a method of computing all almost symmetric generalized numerical semigroup having a fixed Frobenius element and organizing them in a rooted tree depending on a chosen monomial order.

math.CO↗
arXiv2019

Algorithms for Generalized Numerical Semigroups

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in $\mathbb{N}^d$ we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in $\mathbb{N}^d$ we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). It was used to compute the number of such semigroups, and its implementation allowed us to compute (for various dimensions) the number of numerical semigroups for genus that had not been attained before.

math.CO↗
arXiv2019

A generalization of Wilf's conjecture for Generalized Numerical Semigroups

A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in $\mathbb N$. A generalized numerical semigroup is a submonoid of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. In the context of numerical semigroups, Wilf's conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf's conjecture, to the class of $\mathcal C$-semigroups, was proposed by García-García, Marín-Aragón, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf's conjecture, to the setting of generalized numerical semigroups, and prove the conjecture for several large families including the irreducible, symmetric, and monomial case. We also discuss the relationship of our conjecture to the extension proposed by García-García, Marín-Aragón, and Vigneron-Tenorio.

math.CO↗