2024/12/02 by Erkko Lehtonen, Lehtonen, Erkko
Computer Science · #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2412.01107
Extending Sparks's theorem, we determine the cardinality of the lattice of (C1,C2)-clonoids of Boolean functions for certain pairs (C1,C2) of clones of essentially unary, linear, or 0- or 1-separating functions or semilattice operations. When such a (C1,C2)-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 (C1,C2)-clonoids. In the cases when the lattice is finite, we enumerate the corresponding (C1,C2)-clonoids. We also provide a summary of the known results on cardinalities of (C1,C2)-clonoid lattices of Boolean functions.