TITLE: Draw lattice for a join-semillatice
QUESTION [1 upvotes]: I am working with a set that I believe is a join-semilattice (all elements $x,y\in S$ have a least upper bound). I am a little bit confused as to how I would draw a lattice for this set $S=\{1,2,3,12,18,36\}$ when it is partially ordered by divisibility.
My confusion stems from how I would draw the $12$ and $18$ in the lattice. Initially, I would draw a $1$ as the base. To the NW of $1$, I would draw a $2$ with a line going from $1$ to $2$. I would do the same for $1$ and $3$, except I would put the $3$ to the NE of $1$. This is just where I get a tiny bit confused. $2$ and $3$ divide both $12$ and $18$, but $12$ doesn't divide $18$. How would I make this work?

REPLY [4 votes]: Like this rough sketch:

Note that $2$ and $3$ do not have a least upper bound, since $12$ and $18$ are unrelated.
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