Create an order from several incomplete orders
I will answer the following question:


  Given a list of partial orders, how can we find a total order compatible with all of them, if any?


Form a directed graph on the ground set by putting an edge $a \to b$ if $a$ immediately precedes $b$ in one of the partial orders. If this graph contains any directed cycles, then there is no solution. Otherwise, we can find a solution by performing a topological sort.