Płonka Adjunction
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915296743260160 |
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| author | Vidal, Juan Climent Llópez, Enric Cosme |
| author_facet | Vidal, Juan Climent Llópez, Enric Cosme |
| contents | For a signature $Σ$ and its subsignature $Σ^{\neq 0}$ without $0$-ary operation symbols, we prove (1) that there are strong Lawvere adjoint cylinders between the category $\mathsf{Ssl}$, of sup-semilattices, and the categories $\int^{\mathsf{Ssl}}\mathrm{Isys}_Σ$, of sup-semilattice inductive systems of $Σ$-algebras, and $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$, of sup-semilattice inductive systems of $Σ^{\neq 0}$-algebras; (2) that there exists an adjunction between $\mathsf{Ssl}$ and the category $\mathsf{Alg}(Σ^{\neq 0})$, of $Σ^{\neq 0}$-algebras; (3) that there exists an adjunction between the categories $\mathsf{Ssl}$ and $\mathsf{Lnb}$, the category of left normal bands; (4) after defining and stating several technical results on the category $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$, of Płonka $Σ^{\neq 0}$-algebras, and defining functors $J_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$, the tensor product of $\mathsf{Alg}(Σ^{\neq 0})$ and $\mathsf{Lnb}$, and $P_{Σ^{\neq 0}}$ from $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$ to $\mathsf{Alg}(Σ^{\neq 0})$, we prove that $P_{Σ^{\neq 0}}\circ J_{Σ^{\neq 0}}$ has a left adjoint; finally, (5) after defining a functor $\mathrm{Is}_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$ we prove the main result of this paper: that $\mathrm{Is}_{Σ^{\neq 0}}$ has a left adjoint $\mbox{\upshape{Pł}}_{Σ^{\neq 0}}$, which is the Płonka sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03581 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Płonka Adjunction Vidal, Juan Climent Llópez, Enric Cosme Category Theory Primary: 08A62, 08B25, 08C05, 18A40. Secondary: 08A30, 18A30 For a signature $Σ$ and its subsignature $Σ^{\neq 0}$ without $0$-ary operation symbols, we prove (1) that there are strong Lawvere adjoint cylinders between the category $\mathsf{Ssl}$, of sup-semilattices, and the categories $\int^{\mathsf{Ssl}}\mathrm{Isys}_Σ$, of sup-semilattice inductive systems of $Σ$-algebras, and $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$, of sup-semilattice inductive systems of $Σ^{\neq 0}$-algebras; (2) that there exists an adjunction between $\mathsf{Ssl}$ and the category $\mathsf{Alg}(Σ^{\neq 0})$, of $Σ^{\neq 0}$-algebras; (3) that there exists an adjunction between the categories $\mathsf{Ssl}$ and $\mathsf{Lnb}$, the category of left normal bands; (4) after defining and stating several technical results on the category $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$, of Płonka $Σ^{\neq 0}$-algebras, and defining functors $J_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$, the tensor product of $\mathsf{Alg}(Σ^{\neq 0})$ and $\mathsf{Lnb}$, and $P_{Σ^{\neq 0}}$ from $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$ to $\mathsf{Alg}(Σ^{\neq 0})$, we prove that $P_{Σ^{\neq 0}}\circ J_{Σ^{\neq 0}}$ has a left adjoint; finally, (5) after defining a functor $\mathrm{Is}_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$ we prove the main result of this paper: that $\mathrm{Is}_{Σ^{\neq 0}}$ has a left adjoint $\mbox{\upshape{Pł}}_{Σ^{\neq 0}}$, which is the Płonka sum. |
| title | Płonka Adjunction |
| topic | Category Theory Primary: 08A62, 08B25, 08C05, 18A40. Secondary: 08A30, 18A30 |
| url | https://arxiv.org/abs/2305.03581 |