Fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices

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.