Greatest lower bound in a poset

In a partially ordered set, the greatest lower bound of two elements $x$ and $y$ is the "largest" element which is "less than" both $x$ and $y$, in whatever ordering the poset has. In a rare moment of clarity in mathematical naming, the name "greatest lower bound" is a perfect description of the concept: a "lower bound" of two elements $x$ and $y$ is an object which is smaller than both $x$ and $y$ (it "bounds them from below"), and the "greatest lower bound" is the greatest of all the lower bounds.

Formally, if $P$ is a set with partial order $\le$, and given elements $x$ and $y$ of $P$, we say an element $z\in P$ is a lower bound of $x$ and $y$ if $z\le x$ and $z\le y$. We say an element $z\in P$ is the greatest lower bound of $x$ and $y$ if:

• $z$ is a lower bound of $x$ and $y$, and
• for every lower bound $w$ of $x$ and $y$, we have $w\le z$.