\(\varvec{S}\)-preclones and the Galois connection \(\varvec{{}^{S}{}\textrm{Pol}}\)–\(\varvec{{}^{S}{}\textrm{Inv}}\), Part I

We consider S-operations \(f :A^{n} \rightarrow A\) 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 (), and we consider the induced Galois connection \({}^{S}{}\textrm{Pol}\)–\({}^{S}{}\textrm{Inv}\). 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.