Meet in a Preorder

A meet of a family of elements $x_{i}:I\to P$ in a Preorder $P$ is an element $\bigwedge_{i:I}x_{i}:P$ such that

• $\bigwedge_{i:I}x_{i}\leq x_{i}$ for every $i:I$

• For any other $a$ with $a\leq x_{i}$ for every $i:I$, $a\leq\bigwedge_{i:I}x_{i}$

In other words, meets are greatest-lower-bounds of families of elements.

When $I$ is the Booleans, we write $x\land y$ to denote the meet.

Algebraic Presentation

We can present meets purely algebraically, and derive the order after the fact as $x\leq y:=x\land y=y$.