Płonka Adjunction

Abstract

For a signature $\Sigma$ and its subsignature ${\Sigma }^{\ne 0}$ 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 ${\int }^{\mathsf{Ssl}}{\mathrm{Isys}}_{\Sigma }$, of sup-semilattice inductive systems of $\Sigma$-algebras, and ${\int }^{\mathsf{Ssl}}{\mathrm{Isys}}_{{\Sigma }^{\ne 0}}$, of sup-semilattice inductive systems of ${\Sigma }^{\ne 0}$-algebras; (2) that there exists an adjunction between $\mathsf{Ssl}$ and the category $\mathsf{Alg}({\Sigma }^{\ne 0})$, of ${\Sigma }^{\ne 0}$-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 ${\Sigma }^{\ne 0}$-algebras, and defining functors $J_{{\Sigma }^{\ne 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\mathsf{Alg}({\Sigma }^{\ne 0})\otimes \mathsf{Lnb}$, the tensor product of $\mathsf{Alg}({\Sigma }^{\ne 0})$ and $\mathsf{Lnb}$, and $P_{{\Sigma }^{\ne 0}}$ from $\mathsf{Alg}({\Sigma }^{\ne 0})\otimes \mathsf{Lnb}$ to $\mathsf{Alg}({\Sigma }^{\ne 0})$, we prove that $P_{{\Sigma }^{\ne 0}}\circ J_{{\Sigma }^{\ne 0}}$ has a left adjoint; finally, (5) after defining a functor ${\mathrm{Is}}_{{\Sigma }^{\ne 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to ${\int }^{\mathsf{Ssl}}{\mathrm{Isys}}_{{\Sigma }^{\ne 0}}$ we prove the main result of this paper: that ${\mathrm{Is}}_{{\Sigma }^{\ne 0}}$ has a left adjoint $\mbox{\upshape{Pł}}_{Σ^{\neq 0}}$, which is the Płonka sum.

36 pages