Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$.
To determine the maximum value of \( l(\sigma) \) for \( \sigma \in S_n \), we need to find the permutation \(\sigma\) that maximizes the minimum absolute difference between consecutive elements in the permutation. 

1. **Define the problem and notation:**
   - Let \( n \ge 2 \) be a positive integer.
   - \( S_n \) is the set of all permutations of the set \(\{1, 2, \ldots, n\}\).
   - For \(\sigma \in S_n\), define \( l(\sigma) = \min_{1 \le i \le n-1} |\sigma(i+1) - \sigma(i)| \).

2. **Initial observation:**
   - We need to maximize \( l(\sigma) \), which means we want to maximize the smallest difference between consecutive elements in the permutation.

3. **Construct a candidate permutation:**
   - Consider the permutation \(\sigma\) defined by \(\sigma(i) = 2i - 1\) for \(1 \le i \le \left\lceil \frac{n}{2} \right\rceil\) and \(\sigma(i) = 2(i - \left\lceil \frac{n}{2} \right\rceil)\) for \(\left\lceil \frac{n}{2} \right\rceil + 1 \le i \le n\).
   - This permutation alternates between the smallest and largest remaining elements.

4. **Calculate the differences:**
   - For \(1 \le i < \left\lceil \frac{n}{2} \right\rceil\), \(|\sigma(i+1) - \sigma(i)| = 2\).
   - For \(\left\lceil \frac{n}{2} \right\rceil \le i < n\), \(|\sigma(i+1) - \sigma(i)| = 2\).

5. **Verify the minimum difference:**
   - The minimum difference in this permutation is \(2\).

6. **Check if this is the maximum possible:**
   - To see if \(2\) is the maximum possible value for \( l(\sigma) \), consider any other permutation. If there exists a permutation with a minimum difference greater than \(2\), then the differences between consecutive elements would need to be at least \(3\), which is not possible for all pairs in a permutation of \(\{1, 2, \ldots, n\}\).

7. **Conclusion:**
   - Therefore, the maximum value of \( l(\sigma) \) is \(2\).

The final answer is \(\boxed{2}\)