Structured lattices and ground categories of L-sets

Complete lattices are considered with suitable families of lattice morphisms that provide a structure (L,Φ), useful to characterize ground categories of L-sets by means of powerset operators associated to morphisms of these categories. The construction of ground categories and powerset operators presented here extends and unifies most approaches previously considered, allowing the use of noncrisp objects and, with some restriction, the change of base. A sufficiently large category of L-sets that includes all possible ground categories on a structured lattice (L,Φ) is provided and studied, and its usefulness is justified. Many explanatory examples have been given and connection with the categories considered by J. A. Goguen and by S. E. Rodabaugh are stated.