2021/09/10 by Aqsa Bashir, Bashir, Aqsa, Andreas Reinhart +1 · 1 citation
Mathematics · Computer Science · #Rings, Modules, and Algebras #Algebraic structures and combinatorial models #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2109.04764
Let H be a cancellative commutative monoid, let A(H) be the set of atoms of H and let \widetildeH 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 \widetildeH is a DVM, then H is transfer Krull if and only if H⊆\widetildeH is inert. Moreover, we prove that if \widetildeH is factorial, then H is transfer Krull if and only if A(\widetildeH)=\uε| u\inA(H),ε∈\widetildeH×\. We also show that if \widetildeH is half-factorial, then H is transfer Krull if and only if A(H)\subseteqA(\widetildeH). 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.