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Let's consider the following Hasse diagram:

$M_5$

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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2 Answers 2

up vote 4 down vote accepted

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

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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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