The present article aims to develop a categorical duality for the category of bounded complete J-algebraic lattices. In terms of the lattice of weak ideals, we first construct a left adjoint to the forgetful functor Sup \(\rightarrow \) \({\textbf {Pos}}_\vee \) , where Sup is the category of complete lattices and join-preserving maps and \({\textbf {Pos}}_\vee \) is the category of posets and maps that preserve existing binary joins. Based on which, we propose the concept of W-structures over posets and give a W-structure representation for bounded complete J-algebraic posets, which generalizes the representation of algebraic lattices. Finally, we show that the category of join-semilattice WS-structures and homomorphisms is dually equivalent to the category of bounded complete J-algebraic lattices and homomorphisms.
