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 $s' \in Q_j$ for which $s'\in\delta(s,a)$ 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 $Q_1 \leq Q_2 \leq ... \leq Q_n$