# Definition:Ordering Induced by Positivity Property

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## Definition

Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$.

Let $P \subseteq R$ such that:

- $(1): \quad P + P \subseteq P$

- $(2): \quad P \cap \paren {-P} = \set {0_R}$

- $(3): \quad P \circ P \subseteq P$

Then the ordering $\le$ compatible with the ring structure of $R$ is called the **ordering induced by (the positivity property) $P$**.

## Also known as

The **ordering induced by (the positivity property) $P$** is also seen as **ordering defined by (the positivity property) $P$**.

## Also see

This ordering is shown to exist by Positive Elements of Ordered Ring.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.12$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields