Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones

Abstract

Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions for certain pairs $(C_1,C_2)$ of clones of essentially unary, linear, or $0$- or $1$-separating functions or semilattice operations. When such a $(C_1,C_2)$-clonoid lattice is uncountable, the proof is in most cases based on exhibiting a countably infinite family of functions with the property that distinct subsets thereof always generate distinct $(C_1,C_2)$-clonoids. In the cases when the lattice is finite, we enumerate the corresponding $(C_1,C_2)$-clonoids. We also provide a summary of the known results on cardinalities of $(C_1,C_2)$-clonoid lattices of Boolean functions.