I am having some trouble understanding lattices. The following is defined as not being a lattice:
$$S = \{ \{\}, \{y\}, \{x\}, \{x,y,z\}, \{a,x,y,z\}, \{b,x,y,z\} \}$$
I understand that a lattice is a poset for each exists a unique largest element and a unique smallest element (least upper bound and greatest lower bound).
Therefore, is the above not a lattice because we have: $\{a,x,y,z\}$ and $\{b,x,y,z\}$?
In a lattice, for any $\alpha ,\beta$, $\sup (\alpha ,\beta )$ exists.
If your set were to be a lattice, what would be $\sup (\{a,x,y,z\},\{b,x,y,z\})$?
If $\sup (\{a,x,y,z\},\{b,x,y,z\})$ existed in $S$, then there would exist upperbounds for $\{\{a,x,y,z\},\{b,x,y,z\}\}$, but none exist. Therefore $\sup (\{a,x,y,z\},\{b,x,y,z\})$ doesn't exist. Hence $(S,\subseteq )$ is not a lattice.
It is true that $\varnothing$ is the greatest lower bound for $S$ because it is the only lower bound for $S$.