Clonoids of Boolean functions with a linear source clone and a semilattice or 0- or 1-separating target clone

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 Boolean functions. Namely, when $C_1$ is a subclone (a proper subclone, resp.) of the clone of all linear (affine) functions and $C_2$ is a subclone of the clone generated by a semilattice operation and constants (a subclone of the clone of all $0$- or $1$-separating functions, resp.), then the lattice of $(C_1,C_2)$-clonoids is uncountable. Combining this fact with several earlier results, we obtain a complete classification of the cardinalities of the lattices of $(C_1,C_2)$-clonoids for all pairs $(C_1,C_2)$ of clones on $\{0,1\}$.