We describe two problems and their optimal solutions for partially ordered sets. We first describe an optimal algorithm for computing the largest anti-chain of a partially ordered set given its decomposition into its chains. Our algorithm requires $O(n^2m)$ comparisons where $n$ is the number of chains and $m$ is the maximum number of elements in any chain. We also give an adversary argument to prove that this is a lower bound. Our second problem requires us to find if the given poset is a total order. Our optimal algorithm requires $O(mn\log n)$ comparisons. These algorithms have applications in distributed debugging and recovery in distributed systems.