Explanation:

A partially ordered set (poset) is a set equipped with a binary relation that satisfies certain properties. In this case, we have the poset [{1, 2, 3, 4, 6, 9};|], where the set of elements is {1, 2, 3, 4, 6, 9} and the binary relation is denoted by the symbol |.

In order to determine if this poset is a meet semi lattice or a join semi lattice, we need to understand the definitions of these concepts.

Meet Semi Lattice:
A poset is called a meet semi lattice if every pair of elements has a greatest lower bound (meet). In other words, for any two elements a and b in the poset, there exists an element c such that c ≤ a and c ≤ b, and for any other element d that satisfies d ≤ a and d ≤ b, we have d ≤ c.

Join Semi Lattice:
A poset is called a join semi lattice if every pair of elements has a least upper bound (join). In other words, for any two elements a and b in the poset, there exists an element c such that a ≤ c and b ≤ c, and for any other element d that satisfies a ≤ d and b ≤ d, we have c ≤ d.

Analysis:
In the given poset [{1, 2, 3, 4, 6, 9};|], we have the set of elements {1, 2, 3, 4, 6, 9}. Let's analyze the binary relation denoted by |:

- The relation | represents the divisibility relation. For example, 2 | 4 means 2 divides 4.

Now, let's consider the elements 2 and 3 in the poset:

- 2 | 2 since 2 divides itself.
- 2 | 4 since 2 divides 4.
- 2 | 6 since 2 divides 6.
- 2 does not divide 1, 3, or 9.

- 3 | 3 since 3 divides itself.
- 3 does not divide 1, 2, 4, 6, or 9.

From the above analysis, we can see that there is no element c such that c ≤ 2 and c ≤ 3 (greatest lower bound), and there is no element c such that 2 ≤ c and 3 ≤ c (least upper bound). Therefore, the poset [{1, 2, 3, 4, 6, 9};|] does not satisfy the properties of a meet semi lattice or a join semi lattice.

Conclusion:
The correct answer is option 'D' - not a semi lattice.