Clonoids of Boolean functions with a monotone or discriminator source clone

TL;DR

This paper extends Sparks's theorem to determine the size of the lattice of Boolean function clonoids with specific source and target clones, providing explicit descriptions for certain cases involving monotone and discriminator clones.

Contribution

It generalizes Sparks's theorem to new cases and explicitly characterizes clonoids when the source clone is monotone or contains the discriminator function.

Findings

Determined the cardinality of the lattice of (C1,C2)-clonoids with C2 as projections.

Explicitly described (C1,C2)-clonoids for monotone and discriminator source clones.

Extended theoretical understanding of Boolean function clonoids in these cases.

Abstract

Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions in the cases where the target clone $C_2$ is the clone of projections. Moreover, we explicitly describe the $(C_1,C_2)$-clonoids of Boolean functions in the cases where the source clone $C_1$ is one of the four clones of monotone functions or contains the discriminator function.