A characterization of arithmetical invariants by the monoid of relations

The investigation and classification of non-unique factorization phenomena have attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P. Garcia-Sanchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations, and they applied this method successfully in the case of numerical monoids. In this paper, we investigate the algebraic structure of this approach. Thereby, we dispense with the restriction to finitely generated monoids and give applications to other invariants of non-unique factorizations, such as the elasticity and the set of distances

Uniquely presented finitely generated commutative monoids

A finitely generated commutative monoid is uniquely presented if it has only a minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally for some relevant families of numerical semigroups we describe the elements that are uniquely presented.Comment: 13 pages, typos corrected, references update

Presentations of Finitely Generated Cancellative Commutative Monoids and Nonnegative Solutions of Systems of Linear Equations

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143–172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis

Isolated factorizations and their applications in simplicial affine semigroups

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize $\alpha$-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators

Some applications of a new approach to factorization

As highlighted in a series of recent papers by Tringali and the author, fundamental aspects of the classical theory of factorization can be significantly generalized by blending the languages of monoids and preorders. Specifically, the definition of a suitable preorder on a monoid allows for the exploration of decompositions of its elements into (more or less) arbitrary factors. We provide an overview of the principal existence theorems in this new theoretical framework. Furthermore, we showcase additional applications beyond classical factorization, emphasizing its generality. In particular, we recover and refine a classical result by Howie on idempotent factorizations in the full transformation monoid of a finite set