\begin{definition}[Definition:Positivity Property]
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $P \subseteq R$ such that:
{{begin-axiom}}
{{axiom | n = \text P 1
        | m = P + P \subseteq P
}}
{{axiom | n = \text P 2
        | m = P \cap \paren {-P} = \set {0_R}
}}
{{axiom | n = \text P 3
        | m = P \circ P \subseteq P
}}
{{end-axiom}}
Let $\PP: R \to \set {\T, \F}$ be the propositional function defined as:
:$\forall x \in D: \map \PP x \iff x \in P$
Then $\PP$ is the '''positivity property''' on $\struct {R, +, \circ}$.
\end{definition}