On Galois Connections and Soft Computing

Abstract

Abstract. After recalling the different interpretations usually assigned to the term Galois connection, both in the crisp and in the fuzzy case, we survey on several of their applications in Computer Science and, specifically, in Soft Computing.

Generating Isotone Galois Connections on an Unstructured Codomain

“…The following theorem states equivalent definitions of adjunction between posets that can be found in the literature, see for instance [5,16]. …”

Kernels of Residuated Maps as Complete Congruences in Lattices

“…Galois functions were generalized to the fuzzy case by Bělohlávek [14]. Cabrera et al investigate Galois connections in the framework of fuzzy-preordered structures using particular fuzzy equivalence relations with a residuated lattice as the membership-values structure [15][16][17].…”

An algebraic study of LB-valued general fuzzy automata: On the concept of the layers

“…Galois connections originally appeared in the work of Ore [25] to provide a general type of correspondence between structures, and are the generalization of Galois theory introduced by É. Galois to interpret the relationship between field theory and group theory. Galois connections has offered the structure-preserving passage between two worlds of our imagination (cf., Denecke, Erní, and Wismath [10]), and these two mentioned worlds would be so diverse that the least possible connection could be seldom ever imagined (cf., García-Pardo et al, [13]). Moreover, it has been pointed out by Belohalavek [6], that Galois connections capture the very natural rules "the more objects, the less common attributes", and vice-versa.…”

“…These days, Galois connections appear ubiquitary to play a vital role in human reasoning involving hierarchies. For example, some of its applications area covering situations or systems having (i) precise natures are; formal concept analysis (cf., Belohalavek and Konecny [7], Ganter and Wille [12], Wille [35]), category theory (cf., Herrlich and Husek [15], Kerkhoff [21]), logic (cf., Cornejo et al, [9]), category theory, topology and logic (cf., Denecke et al, (Eds) [10]); (ii) imprecise or uncertain natures are; mathematical morphology, category theory (cf., García et al, [14]), fuzzy transform (cf., Perfilieva [27]), Soft computing (cf., García-Pardo et al, [13]); and (iii) vagueness natures; data analysis, reasoning having incomplete information (cf., Järvinen [19]), Pawlak [26], Perfilieva [27]). Here, it is important to note that the equivalence relations based on original Pawlak's (cf., Pawlak [26]), approximation operators form isotone Galois connections and turn out to be interior and closure operators.…”

“…The Galois connections provide the important and fundamental framework to establish interrelationship between different structures involving hierarchies. The Galois connections between two sets, precisely between their power sets equipped with the inclusion order, has two perspectives, namely covariant and contravariant (cf., Birkhoff [8], Erné [10] page 1-138, García et al, [13], Herrlich and Husek [15], Ore [25]). The co-variant Galois connection between two sets is a pair of maps with order-preserving property, and therefore, the term isotone is also used for them.…”