P{\l}onka Adjunction

Abstract

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.