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.

如果对M中的所有\(x, y_ {1},y_{2}\ge 0\)， \(x\le y_ {1}+y_{2} \Rightarrow x=x_{1}+x_{2}\)其中\(0\le x_{i}\le y_{i}\)。我们探索了具有\(M^{+}=\{x\ge 0\mid x\in M\}=M\)的 Riesz 幺半群M生成 Riesz 群的充分必要条件并指出了一些应用。我们称有向 po 幺半群M \(\Pi \) -pre-Riesz 如果\( M^{+}=M\)并且对于所有\(x_{1},x_{2}, \dots ,x_{n }\in M\) , \({{\,\mathrm{glb}\,}}(x_{1},x_{2},\dots ,x_{n})=0\) 或者存在\(r\in \Pi \)使得\(0<r\le x_{1},x_{2},\dots ,x_{n},\)对于某个子集\(\Pi \)的M。我们探讨了\(\Pi \) -pre-Riesz monoids 的示例，以及\(*\) - 不同类型的理想。例如，我们证明如果M是诺特域D的非零（整数）理想的幺半群，并且\(\Pi \)是可逆理想的集合，则M是\(\Pi \) -pre-Riesz 当且仅D是 Dedekind 域。我们还研究了某种类型的前 Riesz 幺半群中的因式分解，并将其与积分域中理想的因式分解理论联系起来。