Mathematics Colloquia and Seminars

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A generalized affine semigroup is an additive submonoid of N^k. As in the case of numerical semigroups, elements of a generalized affine semigroup might have many factorizations into irreducibles (or atoms). Two of the most important arithmetic statistics to measure this phenomenon of non-unique factorization on non-factorial domains/monoids are the system of sets of lengths and the elasticity. In this talk I will target these arithmetic statistics for the family of generalized affine semigroups. The first main goal is to present a partial answer to a conjecture on the sharpness of the Structure Theorem of the Sets of Lengths of Primary Monoids. I will exhibit a generalized affine semigroup with full system of sets of lengths. Then I will construct a generalized affine semigroup that is primary and satisfies almost all requirements of the desired conjecture. On the other hand, it has been conjectured that the elasticity of any generalized affine semigroup is either rational or infinite. In the second part of the talk, I will present a theorem (and if time permits a sketch of its proof) confirming the veracity of such a conjecture.