{
  "id": "2412.01107",
  "version": "v1",
  "published": "2024-12-02T04:27:59.000Z",
  "updated": "2024-12-02T04:27:59.000Z",
  "title": "Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones",
  "authors": [
    "Erkko Lehtonen"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-02T04:27:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean functions",
      "essentially unary",
      "target clones",
      "clonoid lattice",
      "distinct subsets thereof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}