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Non-Unique Factorizations

The package numericalsgps performs computations with and for numerical and affine semigroups. This manuscript is a survey of what the package does, and at the same time intends to gather the trending topics on numerical semigroups.

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numericalsgps, a GAP package for numerical semigroups

Factorization theory: from commutative to noncommutative settings

Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.

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Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

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.

A characterization of arithmetical invariants by the monoid of relations

Let G = Cn1 �···�Cnr with 1 < n1 | ··· |nr be a finite abelian group, d ∗ (G) = n1 +···+nr r, and let d(G) denote the maximal length of a zerosum free sequence over G. Then d(G) � d ∗ (G), and the standing conjecture is that equality holds for G = C r. We show that equality does not hold for C2 � C rn, where n � 3 is odd and r � 4. This gives new information on the structure of extremal zero-sum free sequences over C rn.

On the Davenport constant and on the structure of extremal zero-sum free sequences

The investigation and classification of nonunique factorization phenomena has attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P.A. Garcia-Sanchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations. Then, in Philipp (Semigroup Forum 81:424–434, 2010), the algebraic structure of this approach was investigated and the restriction to finitely generated monoids was removed. We now extend these ideas further to the monotone catenary degree and then apply all these results to the explicit computation of arithmetical invariants of semigroup rings.

A characterization of arithmetical invariants by the monoid of relations II: the monotone catenary degree and applications to semigroup rings

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let D ( G ) be the Davenport constant of G  Then a product of two atoms of H can be written as a product of at most D ( G ) atoms We study this extremal case and consider the set U { 2 , D ( G ) } ( H ) defined as the set of all l ? N with the following property: there are two atoms u , v ? H such that u v can be written as a product of l atoms as well as a product of D ( G ) atoms If G is cyclic, then U { 2 , D ( G ) } ( H ) = { 2 , D ( G ) }  If G has rank two, then we show that (apart from some exceptional cases) U { 2 , D ( G ) } ( H ) = 2 , D ( G ) ? { 3 }  This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank 2 prime power order group uniquely characterizes the group

http://www.diambri.org/Mathpdfs/product-of-2-atoms-ve.pdf

Products of two atoms in Krull monoids and arithmetical characterizations of class groups

Let $G = C_{n_1} \oplus ... \oplus C_{n_r}$ with $1 < n_1 \t ... \t n_r$ be a finite abelian group, $\mathsf d^* (G) = n_1 + ... + n_r - r$, and let $\mathsf d (G)$ denote the maximal length of a zero-sum free sequence over $G$. Then $\mathsf d (G) \ge \mathsf d^* (G)$, and the standing conjecture is that equality holds for $G = C_n^r$. We show that equality does not hold for $C_2 \oplus C_{2n}^r$, where $n \ge 3$ is odd and $r \ge 4$. This gives new information on the structure of extremal zero-sum free sequences over $C_{2n}^r$.

Andreas Philipp

Papers

A characterization of arithmetical invariants by the monoid of relations

On the Davenport constant and on the structure of extremal zero-sum free sequences

A characterization of arithmetical invariants by the monoid of relations II: the monotone catenary degree and applications to semigroup rings

Products of two atoms in Krull monoids and arithmetical characterizations of class groups

On the Davenport constant and on the structure of extremal zero-sum free sequences