Determine whether the lattice below is a complemented lattice:
I'm currently struggling with a problem relating to a lattice that is very similar to the lattice above.
The above lattice is a bounded lattice. However I'm unsure if each element has a complement.
As far as I can tell $e$ doesn't have a complement, therefore not making this a complemented lattice.
$e \land d = b$
$e \land f = c$
You are right, this lattice is not complemented. Since the lattice is relatively small could check this by brute force. That is, for every element $x$ you can check that either $x \wedge e$ is not $a$ (the bottom element) or $x \vee e$ is not $i$ (the top element).
Another approach is to note that $i$ is the only element such that its join with $e$ is the top. That is, here is only one $x$ such that $x \vee e = i$ and that is $x = i$. So if $e$ were to have a complement, then it would have to be $i$. However $e \wedge i = e \neq a$. So $e$ does not have a complement.