#### Presentation Title

Numerical Semigroups: Analysis of Element Factorizations of the same Length

#### Faculty Mentor

Christopher O'Neill

#### Start Date

23-11-2019 8:45 AM

#### End Date

23-11-2019 9:30 AM

#### Location

226

#### Session

poster 2

#### Type of Presentation

Poster

#### Subject Area

physical_mathematical_sciences

#### Abstract

A numerical semigroup is a set of non-negative integers that is closed under addition. Closure under addition means that the sum of any 2 elements from the numerical semigroup is also located in the numerical semigroup. We represent a numerical semigroup by writing it at <6,9,20>, where we call 6,9, and 20 the generators of that numerical semigroup. 6, 9, and 20 are called generators because every element in the numerical semigroup can be represented as the sum of the generators. The research focuses on defining characteristics of numerical semigroups through the analysis of its factorizations and lengths of all factorizations. The factorization of an element are the different ways that an element can be written as a sum of the generators. While the length of a factorization counts how many elements are needed to generate said element. Through the analysis of numerical semigroups with three generators, we developed a function that allows us to determine how many elements display certain lengths which in turn gives us insight into the behavior of its factorizations and lengths.

Keywords: Numerical Semigroup, Factorization, Length

Numerical Semigroups: Analysis of Element Factorizations of the same Length

226

A numerical semigroup is a set of non-negative integers that is closed under addition. Closure under addition means that the sum of any 2 elements from the numerical semigroup is also located in the numerical semigroup. We represent a numerical semigroup by writing it at <6,9,20>, where we call 6,9, and 20 the generators of that numerical semigroup. 6, 9, and 20 are called generators because every element in the numerical semigroup can be represented as the sum of the generators. The research focuses on defining characteristics of numerical semigroups through the analysis of its factorizations and lengths of all factorizations. The factorization of an element are the different ways that an element can be written as a sum of the generators. While the length of a factorization counts how many elements are needed to generate said element. Through the analysis of numerical semigroups with three generators, we developed a function that allows us to determine how many elements display certain lengths which in turn gives us insight into the behavior of its factorizations and lengths.

Keywords: Numerical Semigroup, Factorization, Length