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

We consider S-operationsf:Aⁿ→A in which each argument is assigned a signums∈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-relationsϱ=(ϱs)_s∈S, S-relational clones, and a preservation property (), and we consider the induced Galois connection ^SPol–^SInv. 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.