I'm confused about this definition on wikipedia. Do the GLB and LUB need to be contained in $L$ for it to be a lattice?
if they must be contained, what's the GLB or LUB of two elements $(x,y) \in L$ where $x$ is the LUB or GLB of $L$ itself and happens to be incomparable with $y$? wouldn't the result of that operation need to be outside $L$ if it were to exist? does that mean it's not a lattice?
If the result can't be contained in $L$ then it needs to be outside of $L$, then I guess I would need to define some superset of $L$ that has the same operations as $L$ for the GLB and LUB to live on? in that case is this still a lattice?
The definition on wikipedia says nothing about this.
Let me know if any clarification is needed.