Factorization under local finiteness conditions

Cossu, Laura; Tringali, Salvatore

doi:10.1016/j.jalgebra.2023.04.014

Journal of Algebra

2023

DOI: 10.1016/j.jalgebra.2023.04.014

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Factorization under local finiteness conditions

Laura Cossu

Salvatore Tringali

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Cited by 3 publications

References 53 publications

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Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

Casabella,

D’Anna,

García-Sánchez

2023

Mediterr. J. Math.

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The aim of this article is to study the ideal class monoid $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) of a numerical semigroup S introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) . We observe that $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) is isomorphic to the monoid of ideals of S whose smallest element is 0, which helps to relate $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) to the Apéry sets and the Kunz coordinates of S. We study some combinatorial and algebraic properties of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) , including the reduction number of ideals, and the Hasse diagrams of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) with respect to inclusion and addition. From these diagrams, we can recover some notable invariants of the semigroup. Finally, we prove some results about irreducible elements, atoms, quarks, and primes of $$({\mathscr {C}}\ell (S),+)$$ ( C ℓ ( S ) , + ) . Idempotent ideals coincide with over-semigroups and idempotent quarks correspond to unitary extensions of the semigroup. We show that a numerical semigroup is irreducible if and only if $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) has at most two quarks.

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Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

Casabella,

D’Anna,

García-Sánchez

2023

Mediterr. J. Math.

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On the atomic structure of torsion-free monoids

Gotti,

Vulakh

2023

Semigroup Forum

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Let M be a cancellative and commutative (additive) monoid. The monoid M is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, M satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids.

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On power monoids and their automorphisms

Tringali,

Yan

2025

Journal of Combinatorial Theory, Series A

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Factorization under local finiteness conditions

Cited by 3 publications

References 53 publications

Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

On the atomic structure of torsion-free monoids

On power monoids and their automorphisms

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