  We give a new version of fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata
over distributive lattices: weights are putting in every leaf node of run trees
rather than along with edges from every node to its children. Such settings are
great benefit to obtain complement just by taking dual operation and replacing
each final weight with its complement. We prove that $L$-fuzzy nondeterministic
$\mathrm{B\ddot{u}chi}$ automata have the same expressive power as $L$-fuzzy
alternating $\mathrm{B\ddot{u}chi}$ ones. A direct construction (without
related knowledge about $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ ones
such as: above equivalence relation and their closure properties) is given to
show that the languages recognized by $L$-fuzzy alternating
co-$\mathrm{B\ddot{u}chi}$ automata are also $L$-fuzzy $\omega$-regular.
Furthermore, the closure properties and the discussion about decision problems
for fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata are illustrated in our
paper.
