On transfer Krull monoids

Let H be a cancellative commutative monoid, let A(H) be the set of atoms of H and let (H) over tilde be the root closure of H. Then H is called transfer Krull if there exists a transfer homomorphism from H into a Krull monoid. It is well known that both half-factorial monoids and Krull monoids are transfer Krull monoids. In spite of many examples and counterexamples of transfer Krull monoids (that are neither Krull nor half-factorial), transfer Krull monoids have not been studied systematically (so far) as objects on their own. The main goal of the present paper is to attempt the first in-depth study of transfer Krull monoids. We investigate how the root closure of a monoid can affect the transfer Krull property and under what circumstances transfer Krull monoids have to be half-factorial or Krull. In particular, we show that if (H) over tilde is a DVM, then H is transfer Krull if and only if H subset of (H) over tilde is inert. Moreover, we prove that if (H) over tilde is factorial, then H is transfer Krull if and only if A((H) over tilde) = {u epsilon vertical bar u is an element of A(H), epsilon is an element of (H) over tilde (x)). We also show that if (H) over tilde is half-factorial, then H is transfer Krull if and only if A(H) subset of A((H) over tilde). Finally, we point out that characterizing the transfer Krull property is more intricate for monoids whose root closure is Krull. This is done by providing a series of counterexamples involving reduced affine monoids.