Riesz and pre-Riesz monoids

Abstract

Call a directed partially ordered cancellative divisibility monoid $M$ a Riesz monoid if for all $x,y_{1},y_{2}\geq 0$ in $M,$ $x\leq y_{1}+y_{2}\Rightarrow x=x_{1}+x_{2}$ where $0\leq x_{i}\leq y_{i}$. We explore the necessary and sufficient conditions under which a Riesz monoid $% M $ with $M^{+}=\{x\geq 0|x\in M\}=M$ 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 $x_{1},x_{2},...,x_{n}\in M$, $% glb(x_{1},x_{2},...,x_{n})=0$ or there is $r\in \Pi $ such that $0<r\leq x_{1},x_{2},...,x_{n},$ for some subset $\Pi $ of $M.$ We explore examples of $\Pi $-pre-Riesz monoids of $\star $-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.