Clonoids of Boolean functions with a linear source clone and a semilattice or 0- or 1-separating target clone
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866913788321595392 |
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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 Boolean functions. Namely, when $C_1$ is a subclone (a proper subclone, resp.) of the clone of all linear (affine) functions and $C_2$ is a subclone of the clone generated by a semilattice operation and constants (a subclone of the clone of all $0$- or $1$-separating functions, resp.), then the lattice of $(C_1,C_2)$-clonoids is uncountable. Combining this fact with several earlier results, we obtain a complete classification of the cardinalities of the lattices of $(C_1,C_2)$-clonoids for all pairs $(C_1,C_2)$ of clones on $\{0,1\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04481 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Clonoids of Boolean functions with a linear source clone and a semilattice or 0- or 1-separating target clone 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 Boolean functions. Namely, when $C_1$ is a subclone (a proper subclone, resp.) of the clone of all linear (affine) functions and $C_2$ is a subclone of the clone generated by a semilattice operation and constants (a subclone of the clone of all $0$- or $1$-separating functions, resp.), then the lattice of $(C_1,C_2)$-clonoids is uncountable. Combining this fact with several earlier results, we obtain a complete classification of the cardinalities of the lattices of $(C_1,C_2)$-clonoids for all pairs $(C_1,C_2)$ of clones on $\{0,1\}$. |
| title | Clonoids of Boolean functions with a linear source clone and a semilattice or 0- or 1-separating target clone |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2504.04481 |