Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909411561177088 |
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| author | Lehtonen, Erkko |
| author_facet | Lehtonen, Erkko |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_01107 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones Lehtonen, Erkko Combinatorics Rings and Algebras 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. |
| title | Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2412.01107 |