# Given the Hasse diagram tell if the structure is a lattice

Let's consider the following Hasse diagram:

I need to tell whether this is a lattice. By lattice definition I can prove the above shown structure $M_5$ to be a lattice if and only if $\forall x,y \in M_5$, $\{x,y\}$ has supremum and infimum in $M_5$. Putting all such subsets in a table, not mentioning those subset where $x=y$:

$$\begin{array}{|c || c | c|} \hline Subset & x \wedge y & x \vee y \\ \hline \{a,b\} & b & a \\ \{a,c\} & d & e \\ \{a,d\} & d & a \\ \{a,e\} & a & e \\ \{b,c\} & d & e \\ \{b,d\} & d & b \\ \{b,e\} & b & e \\ \{c,d\} & d & c \\ \{c,e\} & c & e \\ \{d,e\} & d & e \\ \hline \end{array}$$

So the $M_5$ is a lattice.

Is my reasoning in detecting supremum and infimum for each given subset correct? Have I come up with the right conclusion?

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