Definition:Ideal (Order Theory)

A non-empty subset $I$ of an ordered set $\left({P,≤}\right)$ is an ideal, if the following conditions hold:


 * 1) For every $x$ in $I$, $y≤x$ implies that $y \in I$. ($I$ is a lower set)
 * 2) For every $x,\,y \in I$, there is some element $z$ in $I$ such that $x≤z$ and $y≤z$. ($I$ is a directed set)