I am studying about Lattice in Relations and came across this example in a video lecture in which we have to find weather the given Hasse diagram is Join Semi Lattice or not. The Diagram is given below :
When considered the pair {a, b} the upper bounds of these are {c, e, d} as I found and is true as far as I know. My doubt here is that how the Least Upper Bound for them is not c or d but its null ? As observed the points c and d are below e.
The poset with the above Hasse diagram is not a join semilattice (in particular it is not a lattice), because $a$ and $b$ do not have a least upper bound.
By definition a least upper bound of $a$ and $b$ would be a smallest element of the set $S = \{ c,d,e \}$ of common upper bounds of $a$ and $b$.
A smallest element $m \in S$ would in particular need to to satisfy $m \leq c$ and $m \leq d$. But no element of $S$ can satisfy both inequalities.