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Mara Pompili

Publications and source records attributed to Mara Pompili.

7 recordsLinked to original sources
arXiv2026

Factoriality and Class Groups of Upper Cluster Algebras and Finite Laurent Intersection Rings: A Computational Approach

We study factoriality and the class groups of locally acyclic cluster algebras. To do so, we introduce a new class of rings called finite Laurent intersection rings (FLIRs), which includes locally acyclic cluster algebras, full-rank upper cluster algebras, and certain generalized upper cluster algebras and Laurent phenomenon algebras. Our main results are algorithms to compute the class group of an explicit FLIR, to determine factoriality, and to compute all factorizations of a given element. The algorithms are based on multivariate polynomial factorizations, avoiding computationally expensive Gr\"obner basis calculations.

math.AC↗
arXiv2025

On transfer homomorphisms in commutative rings with zero-divisors

We study the arithmetic of monoids of regular elements of commutative rings with zero-divisors. Our focus is on Krull rings and on some of their generalizations (such as weakly Krull rings and C-rings). We establish sufficient conditions for a subring $R$ of a Krull ring $D$ guaranteeing that the inclusion $R^{\bullet} \hookrightarrow D^{\bullet}$ of the respective monoids of regular elements is a transfer homomorphism. The arithmetic of the Krull monoid $D^{\bullet}$ is well studied and the existence of a transfer homomorphism implies that $R^{\bullet}$ and $D^{\bullet}$ share many arithmetic properties.

math.AC↗
arXiv2024

Every finitely generated abelian group is the class group of a generalized cluster algebra

We determine the class group of those generalized cluster algebras that are Krull domains. In particular, this provides a criterion for determining whether or not a generalized cluster algebra is a UFD. In fact, any finitely generated abelian group can be realized as the class group of a generalized cluster algebra. Additionally, we show that generalized cluster algebras are FF-domains and that their cluster variables are strong atoms. Finally, we examine the factorization and ring-theoretic properties of Laurent phenomenon algebras.

math.AC↗
arXiv2023

On class groups of upper cluster algebras

We compute the class group of a full rank upper cluster algebra in terms of its exchange polynomials. As a corollary, we recover a theorem by Cao, Keller, and Qin from 2023 characterizing the UFDs among these algebras. Furthermore, under the additional hypothesis of acyclicity, we obtain a result by Garcia Elsener, Lampe, and Smertnig from 2019. Moreover, we show that all cluster and upper cluster algebras are finite factorization domains, meaning that every non-unit factors as a product of atoms (or irreducibles) and, for each element, there are only finitely many such factorizations up to order and associates. This strengthens another result by Cao, Keller, and Qin showing that cluster and upper cluster algebras are atomic.

math.AC↗
arXiv2023

Ideals and Congruences in L-algebras and Pre-L-algebras

We link the recent theory of $L$-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of $L$-algebras is subtractive and normal in the sense of Zurab Janelidze, but neither the category of $L$-algebras nor that of pre-$L$-algebras are Mal'tsev categories, hence in particular they are not semi-abelian. Therefore $L$-algebras are a rather peculiar example of an algebraic structure.

math.CT↗
arXiv2022

Aspects of the Category SKB of Skew Braces

We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, "product" of ideals of skew braces is often considered. We show that Huq=Smith for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.

math.CT↗