In a (Min) "weak" (or "relaxed") heap, only the right child of each node is guaranteed to have a greater key than the parent node, and the root has no left child (thus, the root is the max element). As in normal heaps, the "heap form" has to be satisfied (i.e., all external nodes have a height difference of at most 1).
If you have access to the ACM Digital Library: Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation
If you have access to Springer books: "Algorithms and Combinatorics" by Korte & Vygen has a nice, clean mathematical description of Fibonacci Heaps, which might also be of interest to you.
