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

Abstract

AbstractWe consider S-operations$$f :A^{n} \rightarrow A$$ f : A n → A in which each argument is assigned a signum$$s \in S$$ s ∈ 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$$\varrho = (\varrho _{s})_{s \in S}$$ ϱ = ( ϱ s ) s ∈ S , S-relational clones, and a preservation property ("Equation missing"), and we consider the induced Galois connection $${}^{S}{}\textrm{Pol}$$ S Pol –$${}^{S}{}\textrm{Inv}$$ S 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.