Generalized (convex) fuzzy sublattices

Following the work of Zadeh and Xue-hai Yuan on the definition of a convex fuzzy subset, three new kinds of definitions of the convex fuzzy sublattices are proposed in this paper. First, by using the relations between fuzzy points and fuzzy sets, the definition of a (β̄,ᾱ)-convex fuzzy sublattice is introduced. The acceptable nontrivial concepts obtained in this manner are the (∈,∈∨q)(∈,∈∨q)-convex fuzzy sublattice and (∈̄,∈̄∨q̄)-convex fuzzy sublattice. Second, the (λ,μ)(λ,μ)-convex fuzzy sublattice is presented and it is also characterized by the level (λ,μ)(λ,μ)-convex sublattice of the given fuzzy set. Finally, (α,β)(α,β)-fuzzy ideal and (∈,∈∨qk)(∈,∈∨qk)-fuzzy sublattice are introduced and several theorems are obtained.