Płonka Adjunction

For a signature $\mathrm{\Sigma }$ and its subsignature ${\mathrm{\Sigma }}^{\mathrm{\ne }0}$ without $0$-ary operation symbols, we prove (1) that there are strong Lawvere adjoint cylinders between the category $\mathsf{S}\mathsf{s}\mathsf{l}$, of sup-semilattices, and the categories ${\int }^{\mathsf{S}\mathsf{s}\mathsf{l}}{\mathrm{I}\mathrm{s}\mathrm{y}\mathrm{s}}_{\mathrm{\Sigma }}$, of sup-semilattice inductive systems of $\mathrm{\Sigma }$-algebras, and ${\int }^{\mathsf{S}\mathsf{s}\mathsf{l}}{\mathrm{I}\mathrm{s}\mathrm{y}\mathrm{s}}_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$, of sup-semilattice inductive systems of ${\mathrm{\Sigma }}^{\mathrm{\ne }0}$-algebras; (2) that there exists an adjunction between $\mathsf{S}\mathsf{s}\mathsf{l}$ and the category $\mathsf{A}\mathsf{l}\mathsf{g}({\mathrm{\Sigma }}^{\mathrm{\ne }0})$, of ${\mathrm{\Sigma }}^{\mathrm{\ne }0}$-algebras; (3) that there exists an adjunction between the categories $\mathsf{S}\mathsf{s}\mathsf{l}$ and $\mathsf{L}\mathsf{n}\mathsf{b}$, the category of left normal bands; (4) after defining and stating several technical results on the category $\text{sffamily}\text{upshape}P\text{ł}Alg({\mathrm{\Sigma }}^{\mathrm{\ne }0})$, of Płonka ${\mathrm{\Sigma }}^{\mathrm{\ne }0}$-algebras, and defining functors $J_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$ from $\text{sffamily}\text{upshape}P\text{ł}Alg({\mathrm{\Sigma }}^{\mathrm{\ne }0})$ to $\mathsf{A}\mathsf{l}\mathsf{g}({\mathrm{\Sigma }}^{\mathrm{\ne }0})\otimes \mathsf{L}\mathsf{n}\mathsf{b}$, the tensor product of $\mathsf{A}\mathsf{l}\mathsf{g}({\mathrm{\Sigma }}^{\mathrm{\ne }0})$ and $\mathsf{L}\mathsf{n}\mathsf{b}$, and $P_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$ from $\mathsf{A}\mathsf{l}\mathsf{g}({\mathrm{\Sigma }}^{\mathrm{\ne }0})\otimes \mathsf{L}\mathsf{n}\mathsf{b}$ to $\mathsf{A}\mathsf{l}\mathsf{g}({\mathrm{\Sigma }}^{\mathrm{\ne }0})$, we prove that $P_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}\circ J_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$ has a left adjoint; finally, (5) after defining a functor ${\mathrm{I}\mathrm{s}}_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$ from $\text{sffamily}\text{upshape}P\text{ł}Alg({\mathrm{\Sigma }}^{\mathrm{\ne }0})$ to ${\int }^{\mathsf{S}\mathsf{s}\mathsf{l}}{\mathrm{I}\mathrm{s}\mathrm{y}\mathrm{s}}_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$ we prove the main result of this paper: that ${\mathrm{I}\mathrm{s}}_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$ has a left adjoint $\text{upshape}{P\text{ł}}_{{\mathrm{\Sigma }}^{\mathrm{\ne }0}}$, which is the Płonka sum.