Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones
Erkko Lehtonen · International Journal of Algebra and Computation · 2025
Extending Sparks’s theorem, we determine the cardinality of the lattice of [Formula: see text]-clonoids of Boolean functions for certain pairs [Formula: see text] of clones of essentially unary, linear, or [Formula: see text]- or [Formula: see text]-separating functions or semilattice operations. When such a [Formula: see text]-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 [Formula: see text]-clonoids. In the cases when the lattice is finite, we enumerate the corresponding [Formula: see text]-clonoids. We also provide a summary of the cardinalities of [Formula: see text]-clonoid lattices of Boolean functions.