S-preclones and the Galois connection SPol–SInv, Part I

We consider S - operations f :Aⁿ → A f : A n → A in which each argument is assigned a signum s ∈ 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 = ( ₛ)s ∈ S ϱ = ( ϱ s ) s ∈ S , S - relational clones , and a preservation property ("Equation missing"), and we consider the induced Galois connection SPol S Pol – SInv 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 .