A Relational Extension of Galois Connections

Relational Galois connections between transitive fuzzy digraphs

“…Given an arbitrary set and a preorder relation defined on , it is possible to lift to the powerset by defining We will use the term powering to refer to the lifting of a preorder to the powerset; thus, , and above are powerings of . Note that the first two relations are actually preorder relations, specifically those used in the construction of the Hoare and Smyth power domains, respectively; the third one neither needs to be reflexive nor transitive, and was introduced in Cabrera et al as a convenient tool to develop relational Galois connections. Furthermore, it is worth noting that the powerings can be defined for any relation not necessarily being a preorder.…”

“…For instance, given two relations and , the ‐Galois condition is given by In Cabrera, we studied the properties of the different extensions obtained in terms of the powerings and used in the corresponding Galois condition. The resulting notion was investigated within the framework of preordered structures in Cabrera et al; later, in another study, we focused our attention on another desirable characterization, the definition of a Galois connection in terms of closures.…”

“…These two papers satisfactorily extend the problem to the fuzzy case in both the domain and range of the Galois connection but, in both cases, the components of the Galois connection are (crisp) functions. A first attempt aiming at obtaining a notion of Galois connection whose components are, in fact, fuzzy functions was given in Cabrera et al; there, we shifted our attention by considering that the domain and range are just sets endowed with arbitrary relations and considering connections whose components are (proper) relations, obtaining what we called relational Galois connections .…”

Towards fuzzy relational Galois connections between fuzzy T-digraphs

“…In [10], we studied the properties of the different extensions obtained in terms of the powerings and used in the corresponding Galois condition. Later, in [7], we focussed our attention on another desirable characterization, the definition of Galois connection in terms of closures. We introduce below the corresponding relational extension of the notion of closure operator.…”

“…Our first attempt to obtain a properly fuzzy notion of Galois connection was to go back to the crisp case and consider a suitable relational generalization of Galois connection, one in which the domain and range are just sets endowed with arbitrary relations and whose components are (proper) relations, and this is the content of [7]. It is worth noting that, in order to preserve the existing construction via closures, we needed to provide a relational definition of Galois connection which allows the composition of the two components of the connection, and this is something that is not guaranteed by some of the relational extensions of Galois connection that can be found in the literature.…”