Płonka adjunction

Let $\varSigma $ be a signature without $0$-ary operation symbols and $\textsf{Sl}$ the category of semilattices. Then, after defining and investigating the categories $\int <^>{\textsf{Sl}}\textrm{Isys}_{\varSigma }$, of inductive systems of $\varSigma $-algebras over all semilattices, which are ordered pairs $\mathscr{A}= (\textbf{I},\mathscr{A})$ where $\textbf{I}$ is a semilattice and $\mathscr{A}$ an inductive system of $\varSigma $-algebras relative to $\textbf{I}$, and P & lstrok;Alg$ (\varSigma )$, of P & lstrok;onka $\varSigma $-algebras, which are ordered pairs $(\textbf{A},D)$ where $\textbf{A}$ is a $\varSigma $-algebra and $D$ a P & lstrok;onka operator for $\textbf{A}$, i.e. in the traditional terminology, a partition function for $\textbf{A}$, we prove the main result of the paper: There exists an adjunction, which we call the P & lstrok;onka adjunction, from $\int <^>{\textsf{Sl}}\textrm{Isys}_{\varSigma }$ to P & lstrok;Alg$ (\varSigma )$.