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