# Is there a commonly used name for this weakening of the Archimedean property?

Suppose that $$R$$ is a partially ordered commutative ring. The order behaves nicely: We have $$0 \leq 1$$. If $$a \leq b$$ and $$c > 0$$, then $$a + c \leq b + c$$ and $$a \cdot c \leq b \cdot c$$. For $$n \in \mathbb{Z}$$ define $$\hat{n} = 1 + \cdots + 1$$(n terms), where $$1$$ is the ring's multiplicative identity. The property of the ring is:

For all $$r \in R$$ such that $$r > 0$$ is an $$n \in \mathbb{Z}$$ such that either $$1 < \hat{n} \cdot r$$ or $$\hat{n} \cdot r \nleq 1$$.

The first case is like the Archimedean property. The second case says there are none of the usual infinitesimals.

If this property has a commonly used name, I would like to know it. Any references would be appreciated.

• I am not sure what Exactly you want here. When we have either "$(1 \lt \hat{n} \cdot r)$" or "$(\hat{n} \cdot r \not \leq 1)\equiv(1 \lt \hat{n} \cdot r)$" , we have the Same Equivalent Statement.
– Prem
May 10, 2023 at 18:02
• The order is a partial order not a total order.
– Jay
May 10, 2023 at 18:36
• One Statement says , we can multiply a number such that we get a number larger than 1. Other Statement says , we can multiply a number such that we get a number which is not less than 1 , which means it is larger than 1. Seems Equivalent , though I might be wrong. Thinking !
– Prem
May 10, 2023 at 18:50
• @Prem In a partial order items can be incomparable.
– Jay
May 10, 2023 at 18:53
• I realize that , but I see "$\le$" , "$\gt$" , "$\not \le$" , where it is not clear which is the Partial Order. In Case that is a valid objection (I am not sure) , maybe you could edit it such that you use only that Partial Order through-out. In Case that is not a valid objection , you can leave it unchanged & maybe somebody will Post some Answer !
– Prem
May 10, 2023 at 19:16