Riesz and pre-Riesz monoids

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 $$ with $M^{+}=\{x\geq 0|x\in M\}=M$ generates a Riesz group and indicate some applications. We call a directed p.o. monoid $M$ $\Pi$-pre-Riesz if $$ and for all $x_{1},x_{2},...,x_{n}\in M$, $$ 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.