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We give a new version of fuzzy alternating $\mathrm{B\ddot{u}chi}$ automataover distributive lattices: weights are putting in every leaf node of run treesrather than along with edges from every node to its children. Such settings aregreat benefit to obtain complement just by taking dual operation and replacingeach final weight with its complement. We prove that $L$-fuzzy nondeterministic$\mathrm{B\ddot{u}chi}$ automata have the same expressive power as $L$-fuzzyalternating $\mathrm{B\ddot{u}chi}$ ones. A direct construction (withoutrelated knowledge about $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ onessuch as: above equivalence relation and their closure properties) is given toshow that the languages recognized by $L$-fuzzy alternatingco-$\mathrm{B\ddot{u}chi}$ automata are also $L$-fuzzy $\omega$-regular.Furthermore, the closure properties and the discussion about decision problemsfor fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata are illustrated in ourpaper.

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