A Unified View of Different Axiomatic Measures Defined on $L$-Fuzzy Sets

The recent literature contains a multitude of extensions of (axiomatic) notions from the context of ordinary fuzzy sets to more general contexts. Using the language of lattices, we provide a general and compact formulation encompassing a large number of those notions and their potential extensions to even more complex frameworks. The new formulation offers a unifying perspective of the different measures and operations between (generalized) fuzzy sets, and has a potential impact on the simplification of the redundant mathematical proofs concerning the formal relations between the different notions and the properties of certain particular constructive definitions.