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.