Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

By Yushuang Fan, Alfred Geroldinger, Florian Kainrath, S. Tringali. 2016

47 citations recorded by the source corpus.

Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \in \mathbb N$, let $\mathscr U_k (H)$ denote the set of all $\ell \in \mathbb N$ with the property that there are atoms $u_1, \ldots, u_k, v_1, \ldots, v_{\ell}$ such that $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$ (thus, $\mathscr U_k (H)$ is the union of all sets of lengths containing $k$). The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets $\mathscr U_k (H)$ are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/86a40fff51a18c90a6b3e1131bf464ef4e311f07

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In Algebra and Number Theory

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Citations in the corpus

Factorization invariants of Puiseux monoids generated by geometric sequences (2019, background): “e set λk(M) := min Uk(M) and ρk(M) := sup Uk(M), andwe call ρk(M)the k-th local elasticity ofM. Unions ofsets oflengths have received a great deal of attention in recent literature; see, for example, [4, 7, 15, 43]. In particular, the unions of sets of lengths and the local elasticities of Puiseux m”
On strongly primary monoids, with a focus on Puiseux monoids (2019, background): “The structure of unions of sets of lengths has been studied for a wide range of monoids and domains (see [14, 4, 15, 55] for recent progress).”
Power monoids: A bridge between factorization theory and arithmetic combinatorics (2017, background): “On the one hand, there has been a mounting interest for possible generalizations of factorization theory to monoid-like structures that need no longer be commutative or cancellative [5, 13, 17, 24, 25, 29, 52].”