On the Existence of Isotone Galois Connections between Preorders

Relational Galois connections between transitive fuzzy digraphs

“…In this paper, we continue our research line on the construction of Galois connections between sets with unbalanced structures initiated in Garcia‐Pardo et al, where we attempted to characterize the existence of the right part of a Galois connection of a given mapping between sets with a different structure (it is precisely this condition of different structure that makes this problem to be outside the scope of Freyd's adjoint functor theorem). Since then, we have obtained results in several frameworks: for instance, in Garcia‐Pardo et al, given a mapping from a (pre‐)ordered set into an unstructured set , we solved the problem of completing the structure of , namely, defining a suitable (pre‐)ordering relation on , such that there exists a mapping such that the pair of mappings forms an isotone Galois connection (or adjunction) between the (pre‐)ordered sets and .…”