On transfer Krull monoids

Abstract

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 $$\widetilde{H}$$ <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 $$\widetilde{H}$$ <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 $$H\subseteq \widetilde{H}$$ <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 $$\widetilde{H}$$ <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 $$\mathcal {A}(\widetilde{H})=\{u\varepsilon \mid u\in \mathcal {A}(H),\varepsilon \in \widetilde{H}^{\times }\}$$ <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 $$\widetilde{H}$$ <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 $$\mathcal {A}(H)\subseteq \mathcal {A}(\widetilde{H})$$ <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.

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