# 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.