# 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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thanks for asking such a good question, helped me a lot. :D –  Hassan May 26 at 15:21
for {a,d} can we say that avd can also be e? –  Hassan May 26 at 15:29

Your calculation of the supremum and infimum is correct and the structure is a lattice.

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for {a,d} can we say that avd can also be e? –  Hassan May 26 at 15:29

An alternate and shorter way would be to check if the meet operation holds for each element and if the lattice is bounded above, which it is. This is in reference to the fact that every meet-lattice with a greatest element is a join lattice.

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