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

TL;DR

This paper extends Sparks's theorem to determine the size and structure of lattices of Boolean function clonoids for specific clone pairs, providing enumeration and cardinality results.

Contribution

It generalizes the classification of Boolean clonoid lattices for pairs of clones with specific properties, including enumeration and cardinality analysis.

Findings

Identifies when the clonoid lattice is finite or uncountable.

Provides enumeration of all clonoids in finite cases.

Establishes methods for constructing infinite families of functions for uncountable lattices.

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.