Sets of Lengths in Krull monoids

We all do know from highschool the Fundamental Theorem of Arithmetic: Every positive integer can be factorized in a unique way as a product of primes. Prime numbers are (multiplicatively) indecomposable numbers (also called atoms), such as 2, 3, 5, 7, 11, 13, 17, 19, .... The study of prime numbers is a classical subfield of Pure Mathematics, whose results are indispensable in coding theory and cryptography. There are many domains in mathematics, where as in the positive integers the objects (elements) of the domain can be decomposed into atoms. However, in general such factorizations (decompositions) into atoms need not be unique. When talking about such domains we have in mind more general domains of numbers (keyword: algebraic integers), or domains of functions, or generalized vector spaces (every finite dimensional vector space can be decomposed into one- dimensional vector spaces; thus in this case the atoms are the one dimensional vector spaces). In the present project we study very general domains, so-called Krull monoids with finite class group. Factorizations in Krull monoids are unique as in the positive integers if and only if the class group consists of precisely one element. However in general, class groups can be arbitrarily large. If an element a of a Krull monoid H has a factorization into k atoms, then k is called the length of the factorization and L(a) denotes the set of all possible factorization lengths of a. All sets of lengths are finite, but there are arbitrarily large sets of lengths. Their structure provides information how elements are decomposed into atoms. For a given Krull monoid H we study the (infinite) system L(H) = {L(a) | a H} of sets of lengths of all elements. This will be done with methods from Additive Combinatorics. 1

We all do know from highschool the 'Fundamental Theorem of Arithmetic': It states that Every positive integer can be factorized in a unique way as a product of primes. Prime numbers are (multiplicatively) indecomposable numbers (also called atoms), such as 2, 3, 5, 7, 11, 13, 17, and 19. The study of prime numbers is a classical subfield of Pure Mathematics, whose results are indispensable in coding theory and cryptography. There are many domains in mathematics, where, as in the positive integers, the objects (elements) of the domain can be decomposed into atoms. However, in general, such factorizations (decompositions) into atoms need not be unique. When talking about such domains we have in mind more general domains of numbers (keyword: algebraic integers), or domains of functions, or generalized vector spaces (every finite-dimensional vector space can be decomposed into one-dimensional vector spaces; thus, in this case, the atoms are the one-dimensional vector spaces). In the present project, we studied very general domains, called Krull monoids with finite class group. Factorizations in Krull monoids are unique -- as in the positive integers -- if and only if the class group consists of precisely one element. However, in general, class groups can be arbitrarily large. If an element x of a Krull monoid has a factorization into k atoms, then k is called the length of the factorization and the set of all possible factorization lengths of this element is called the set of lengths of x. All sets of lengths are finite, but there are arbitrarily large sets of lengths. Their structure provides information how elements are decomposed into atoms. In our project, we used modern methods from algebra, number theory, and combinatorics. Among others, we characterized those Krull monoids whose sets of lengths are arithmetical progressions.