‘Level shifting’
Given any set \(S\), there is a set \(\mathbf{Rel}(S)\) of binary relations on \(S\) (i.e. \(\mathbb{P}(S \times S)\))
This power set can be given a preorder structure using the subset relation.
A subset of possible relations satisfy the axioms of preorder relations. \(\mathbf{Pos}(S) \subseteq \mathbb{P}(S \times S)\) which again inherits a preorder structure from the subset relation
A preorder on the possible preorder structures of \(S\), that’s a level shift.
The inclusion map \(\mathbf{Pos}(S) \hookrightarrow \mathbf{Rel}(S)\) is a right adjoint to a Galois connection, while its left adjoint is \(\mathbf{Rel}(S)\overset{Cl}{\twoheadrightarrow} \mathbf{Pos}(S)\) which takes the reflexive and transitive closure of an arbitrary relation.