Definition:Ordering Induced by Positivity Property

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.

• Definition:Total Ordering Induced by Strict Positivity Property‎

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