The following is an excerpt from my textbook:
A lattice need not be a totally ordered set. Consider the partially ordered set $(ω,D)$ where D is the relation on $ω$ defined by $x D y$ iff $x | y$. Then $(ω,D)$ is a lattice. For all $x,y \in ω$, $inf${$x,y$} is the highest common factor of $x$ and $y$ and $sup${$x,y$} is the lowest common multiple of $x$ and $y$. For instance, $inf${$4,6$}$=2$ and $sup${$4,6$}$=12$. However, D is not a total order relation.
I'm confused about $inf${$4,6$}$=2$. In the context of the example, my understanding is that the greatest lower bound of the set {$4,6$} is some element of $ω$ such that $ω \le 4,6$. Wouldn't $4$ fit that definition and not 2?
No, because $4 \not \leq 6$ by the relation we are using, which is divisibility. $4$ does not divide into $6$ evenly. The $\inf$ of $4$ and $6$ is the largest number which divides them both, the $\gcd$, which is $2$