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Riesz and pre-Riesz monoids

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Call a directed partially ordered cancellative divisibility monoid M a Riesz monoid if for all \(x,y_{1},y_{2}\ge 0\) in M,  \(x\le y_{1}+y_{2}\Rightarrow x=x_{1}+x_{2}\) where \(0\le x_{i}\le y_{i}\). We explore the necessary and sufficient conditions under which a Riesz monoid M with \(M^{+}=\{x\ge 0\mid x\in M\}=M\) generates a Riesz group and indicate some applications. We call a directed p.o. monoid M \(\Pi \)-pre-Riesz if \( M^{+}=M\) and for all \(x_{1},x_{2}, \dots ,x_{n}\in M\), \({{\,\mathrm{glb}\,}}(x_{1},x_{2},\dots ,x_{n})=0\) or there is \(r\in \Pi \) such that \(0<r\le x_{1},x_{2},\dots ,x_{n},\) for some subset \(\Pi \) of M. We explore examples of \(\Pi \)-pre-Riesz monoids of \(*\)-ideals of different types. We show for instance that if M is the monoid of nonzero (integral) ideals of a Noetherian domain D and \(\Pi \) the set of invertible ideals, M is \(\Pi \)-pre-Riesz if and only D is a Dedekind domain. We also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain.

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Acknowledgements

I am indebted to Brian Davey, the managing editor of Algebra Universalis, for straightening my original write up to make it look like a decent paper. In addition I must admit that I did not work in a vacuum, a lot of folks helped and I am thankful to all.

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  1. Department of Mathematics, Idaho State University, Pocatello, ID , 83209, USA

    Muhammad Zafrullah

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  1. Muhammad Zafrullah
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Correspondence to Muhammad Zafrullah.

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Zafrullah, M. Riesz and pre-Riesz monoids. Algebra Univers. 83, 9 (2022). https://doi.org/10.1007/s00012-021-00765-y

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