
TL;DR
This paper investigates the structure of Riesz and pre-Riesz monoids, establishing conditions for their relation to Riesz groups and exploring applications in ideal theory of domains.
Contribution
It characterizes when Riesz monoids generate Riesz groups and analyzes pre-Riesz monoids, especially in the context of ideal structures in Noetherian and Dedekind domains.
Findings
Riesz monoids generate Riesz groups under specific conditions.
Characterization of $ ext{Pi}$-pre-Riesz monoids in ideal theory.
Connection between factorization in pre-Riesz monoids and ideal factorization.
Abstract
Call a directed partially ordered cancellative divisibility monoid a Riesz monoid if for all in where . We explore the necessary and sufficient conditions under which a Riesz monoid with generates a Riesz group and indicate some applications. We call a directed p.o. monoid -pre-Riesz if and for all , or there is such that for some subset of We explore examples of -pre-Riesz monoids of -ideals of different types. We show for instance that if is the monoid of nonzero (integral) ideals of a Noetherian domain and the set of invertible ideals, is -pre-Riesz if and only is a…
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