On transfer Krull monoids

Let $H$ be a cancellative commutative monoid, let $mathcal{A}(H)$ be the set of atoms of $H$ and let $widetilde{H}$ 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 counter examples 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 $widetilde{H}$ is a DVM, then $H$ is transfer Krull if and only if $Hsubseteqwidetilde{H}$ is inert. Moreover, we prove that if $widetilde{H}$ is factorial, then $H$ is transfer Krull if and only if $mathcal{A}(widetilde{H})={uvarepsilonmid uinmathcal{A}(H),varepsiloninwidetilde{H}^{times}}$. We also show that if $widetilde{H}$ is half-factorial, then $H$ is transfer Krull if and only if $mathcal{A}(H)subseteqmathcal{A}(widetilde{H})$. 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.