2022/06/14 by Aqsa Bashir, Andreas Reinhart
Computer Science · Mathematics · #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.1007/s00233-022-10296-0
crossref issued 2022/06/14 · crossref published 2022/06/14 · crossref published-online 2022/06/14 · openalex publication_date 2022/06/14 · crossref created 2022/06/14 · crossref published-print 2022/08/01 · crossref deposited 2022/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03 · crossref indexed 2026/08/03
Abstract Let H be a cancellative commutative monoid, let \mathcal A(H) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> be the set of atoms of H and let \widetildeH <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> 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 \widetildeH <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> is a DVM, then H is transfer Krull if and only if H⊆ \widetildeH <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>⊆</mml:mo> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> </mml:math> is inert. Moreover, we prove that if \widetildeH <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> is factorial, then H is transfer Krull if and only if \mathcal A(\widetildeH)=\uε | u∈ \mathcal A(H),ε ∈ \widetildeH× \ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo></mml:mo> <mml:mi>u</mml:mi> <mml:mi>ε</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>×</mml:mo> </mml:msup> <mml:mo></mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We also show that if \widetildeH <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> is half-factorial, then H is transfer Krull if and only if \mathcal A(H)⊆ \mathcal A(\widetildeH) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊆</mml:mo> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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.