Fuzzy Galois connections categorically

Abstract

Key words Unital quantale, category enriched over a unital quantale, Ω-adjunction, Ω-relation, covariant fuzzy Galois connection, contravariant fuzzy Galois connection, Girard quantale. MSC (2000)03B52, 06A15, 06F07, 18D20 This paper presents a systematic investigation of fuzzy (non-commutative) Galois connections in the sense of R. Bělohlávek [1], from the point of view of enriched category theory. The results obtained show that the theory of enriched categories makes it possible to present the theory of fuzzy Galois connections in a succinct way; and more importantly, it provides a useful method to express and to study the link and the difference between the commutative and the non-commutative worlds.

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

“…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.…”

Fuzzy Connections and Relations in Complete Residuated Lattices

“…A Galois connection is an important mathematical tool for algebraic structure, data analysis, and knowledge processing [1][2][3][4][5][6][7][8][9][10]. Hájek [11] introduced a complete residuated lattice L that is an algebraic structure for many-valued logic.…”