Consider the set $X = \{a,b,c,d,e\}$ under the partial ordering $$R = \{(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)\}.$$ The Hasse diagram of the partial order $(X,R)$ is shown below. The minimum number of ordered pairs that need to be added to $R$ to make $(X,R)$ a lattice is _________ .

A partially ordered set $(X, R)$ is a lattice if every pair of elements has a unique least upper bound (LUB / join, $\vee$) and a unique greatest lower bound (GLB / meet, $\wedge$).

Let us verify whether every pair of elements in $(X, R)$ already has a unique LUB and GLB:

1. Pairs involving $a$ or $e$:

   • $a$ is the least element (bottom), so for any $x \in X$, $a \wedge x = a$ and $a \vee x = x$.

   • $e$ is the greatest element (top), so for any $x \in X$, $e \wedge x = x$ and $e \vee x = e$.

2. Pair $\{b, c\}$:

   • Since $b \le c$, $b \wedge c = b$ and $b \vee c = c$.

3. Pair $\{b, d\}$:

   • Lower bounds of $\{b, d\}$: $\{a\} \implies b \wedge d = a$.

   • Upper bounds of $\{b, d\}$: $\{e\} \implies b \vee d = e$.

4. Pair $\{c, d\}$:

   • Lower bounds of $\{c, d\}$: $\{a\} \implies c \wedge d = a$.

   • Upper bounds of $\{c, d\}$: $\{e\} \implies c \vee d = e$.

Since every pair of elements in $X$ already has a unique GLB and a unique LUB, the given poset $(X, R)$ is already a lattice.

Therefore, the minimum number of ordered pairs that need to be added is 0.