How to Represent ๐ด โ ๐ต โ โข โข โข โ ๐: From Curried Functions and Hyperfunctions to Curried Structures and Hyperstructures, and more
Toshio Fujita ยท 2025
Curried functions transform multi-argument functions into nested single-argument functions, enabling partial application and undergirding many features of functional programming. Their generalization, curried ๐-ary functions, intuitively extends the usual ๐ด โ ๐ต mapping to ๐ด โ ๐ต โ ๐ถ and, more generally, to ๐ด โ ๐ต โ โข โข โข โ ๐. Meanwhile, the notions of Hyperfunction and ๐-Superhyperfunction extend classical functions by mapping elements into (iterated) power sets, thereby capturing hierarchical function structures [1, 2]. Although both curried ๐-ary functions and Hyperfunctions (and ๐-Superhyperfunctions) have rich mathematical properties and important applications in programming, their fusion has not yet been explored. To fill this gap, we introduce the concepts of Curried Hyperfunction and Curried Superhyperfunction, and further define Curried Structure, Curried Hyperstructure, and Curried SuperHyperstructure. We also investigate curried-structure analogues in fuzzy sets and logical systems. Finally, we analyze these constructions' algebraic properties, establish their interrelationships, and discuss potential applications in both mathematical theory and computational practice.