Clonoids of Boolean functions with a linear source clone and a semilattice or 0- or 1-separating target clone
Erkko Lehtonen
TL;DR
This work extends Sparks's theorem by classifying the cardinalities of the clonoid lattice $\,\mathcal{L}_{(C_1,C_2)}$ for Boolean functions when the source clone $C_1$ is linear and the target clone $C_2$ lies in semilattice- or $0$-/1-separating-structured intervals. The authors prove uncountability of the lattice for key pairs and construct a countable family of Boolean functions such that distinct subsets generate distinct clonoids, yielding a continuum of clonoids. They then establish a complete classification of all clone pairs on $\\{0,1\}$ according to whether the corresponding clonoid lattice is finite, countable, or uncountable, combining new uncountability results with extensive prior work. A complementary linear-algebra perspective ties certain clonoids to affine-subspace configurations, while monotonicity provides a way to derive additional clonoids from known superclones. The results enhance understanding of clonoids, with implications for the algebraic analysis of CSPs and the structure of clone lattices, and outline open directions for generalization beyond the Boolean setting.
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 $\{0,1\}$.