Extension of a partial order to a total of partitions of a weak alternating automaton My problem is this: given a weak alternating automaton and its partitions, and given a partial order on these partitions, how do we extend the partial order to a total order?

The partitions of weak alternating automaton are defined as follows: Let $A(\Sigma,Q,q_0,\delta,\alpha)$ where $\alpha$ represents the acceptance condition, $Q$ is the set of states, and $q_0$ is the initial state. A weak alternating automaton is an alternating Buchi tree automaton in which exists a partition of the set $Q$ into disjoint sets $Q_1,\dotsc,Q_n$, such that for each set $Q_i$, either $Q_i\subseteq\alpha$ (in which case $Q_i$ is an accepting set) or $Q_i\cap\alpha=\varnothing$ (in which case $Q_i$ is a rejecting set). In addition exists a partial order $\leq$ on the collection of $Q_i$, such that for every $s \in Q_i$ and $s&#39; \in Q_j$ for which $s&#39;\in\delta(s,a)$ we have $Q_j \leq Q_i$. Thus, transition from the state in $Q_i$ lead to states in either the same $Q_i$ or lower one.Hence , given the partial order defined above, how to turn this order in a total order $Q_1 \leq Q_2 \leq ... \leq Q_n$
If I understand you correctly, you are asking for a linear extension of the partial order induced by the partitioning of the weak alternating automaton into its maximal strictly connected components and the transitions between the states. 

Note that the "extension of the partial order" to a total order is not unique. The process of obtaining any such total order is called topologic sorting.

See the following wikipedia page for details and a link to an algorithm: http://en.wikipedia.org/wiki/Partially_ordered_set