Abstract: The interval sets arc new important direction of the research, which have been widely applied in approximate reasoning,fuzzy control, and so on. In the interval sets, the new interval implication (→) was redefined, the lattice implication algebras was reconstructed and its properties were discussed. Meanwhile, commutative FI-algebras and MV-algebras also were redefined in the interval sets. Three different algebraic systems were proved to equivalence, which are lattice implication algebras, commutative FI-algebras and MV-algebras.

Key words: Interval sets,Lattice implication algebras,FI-algebras,MV-algebras