0. Some definitions

In any partially ordered set $P$,

1. $u$ is an upper bound of $x$ and $y$ if $x \leq u$ and $y \leq u$.

2. $z$ is a least upper bound of $x$ and $y$ if
   1. $z$ is an upper bound of $x$ and $y$, and

   2. $z \leq u$ for all upper bounds $u$ of $x$ and $y$.

   We also can say that $z$ is the join of $x$ and $y$. We write $z$ = lub$(x,y)$ or $z = x \vee y$.

3. $z$ is a greatest lower bound of $x$ and $y$ if ...We also say $z$ is the meet of $x$ and $y$ and write $z =$   glb$(x,y)$ or $z = x \wedge y$.