Let's break down the problem.

We are given a set $P = \{1, 2, 3, 4, 5\}$ and a partial ordering $R$ consisting of the following relations:

$R = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 5), (3, 3), (3, 5), (4, 4), (4, 5), (5, 5)\}$

Key Definitions:

1. Partial Ordering: A partial order is a binary relation that is reflexive, antisymmetric, and transitive.
2. Lattice: A lattice is a partially ordered set in which every pair of elements has both a least upper bound (join) and a greatest lower bound (meet).

Step-by-step analysis:

1. Check the partial order properties:

   • Reflexivity: Each element is related to itself, as we see $(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)$.
   • Antisymmetry: If $a \leq b$ and $b \leq a$, then $a = b$. The given pairs comply with this.
   • Transitivity: If $a \leq b$ and $b \leq c$, then $a \leq c$. Based on the given pairs, this holds (e.g., $1 \leq 2$ and $2 \leq 3$ implies $1 \leq 3$).

2. Check if the relation defines a lattice:

   • Greatest Lower Bound (GLB) or meet: For every pair of elements $a$ and $b$, there should be a greatest element $c$ such that $c \leq a$ and $c \leq b$.
   • Least Upper Bound (LUB) or join: For every pair of elements $a$ and $b$, there should be a least element $d$ such that $a \leq d$ and $b \leq d$.

To form a lattice, we need to ensure that for every pair of elements in the set $P$, both the meet and join exist.

3. Examine the structure: The given ordering of elements seems to already define a structure where:
   • 1 is the least element (since $1 \leq 2$, $1 \leq 3$, $1 \leq 4$, $1 \leq 5$).
   • 5 is the greatest element (since $2 \leq 5$, $3 \leq 5$, $4 \leq 5$).
   • Intermediate relationships such as $2 \leq 3 \leq 5$ and $4 \leq 5$ further support this.

Thus, this structure already allows for the meet and join for any pair of elements. Therefore, no additional pairs need to be added.

Conclusion:

The minimum number of ordered pairs that need to be added to $R$ to make $(P, R)$ a lattice is indeed zero. This means that $(P, R)$ is already a lattice.

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Would you like more details or have any questions?
Here are 5 related questions to explore further:

1. What is the difference between a partial order and a total order?
2. How can we construct the Hasse diagram of the given relation $R$?
3. What conditions must a partially ordered set satisfy to be considered a lattice?
4. Can a lattice be finite or infinite, and how does that affect its structure?
5. How do meet and join operations behave in lattices with more complex sets?

Tip: Always verify reflexivity, antisymmetry, and transitivity when checking if a relation is a partial order.