$S$-preclones and the Galois connection ${}^S\mathrm{P}\mathrm{o}\mathrm{l}$-${}^S\mathrm{I}\mathrm{n}\mathrm{v}$, Part I

We consider $S$-operations $f\text{\hspace{0.17em}⁣}:A^n\to A$ in which each argument is assigned a signum $s\in S$ representing a “property” such as being order-preserving or order-reversing with respect to a fixed partial order on $A$. The set $S$ of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of $S$-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all $S$-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of $S$-preclone. We introduce $S$-relations $\varrho =({\varrho }_s{)}_{s\in S}$, $S$-relational clones, and a preservation property ($f\varrho$), and we consider the induced Galois connection ${}^S\mathrm{P}\mathrm{o}\mathrm{l}$-${}^S\mathrm{I}\mathrm{n}\mathrm{v}$. The $S$-preclones and $S$-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all $S$-preclones on $A$.