In this paper, properties of operations and algebraic structures of hesitant fuzzy sets are investigated. Semilattices of hesitant fuzzy sets with union and intersection are discussed, respectively. By using ⊕ and ⊗ operators, the commutative monoid of hesitant fuzzy sets is provided, moreover, the lattice and distributive lattice of hesitant fuzzy sets are defined on the equivalence class of hesitant fuzzy sets. Based on the distributive lattice of hesitant fuzzy sets, the residuated lattices of hesitant fuzzy sets are constructed by residual implications, which are induced by intersection and ⊗, respectively. From the theoretical point of view, algebraic structures of hesitant fuzzy sets are useful for approximate reasoning and decision making to deal with hesitancy of information.