Multi-adjoint concept lattices via quantaloid-enriched categories

Abstract

With quantaloids carefully constructed from multi-adjoint frames, it is shown that multi-adjoint concept lattices, multiadjoint property-oriented concept lattices and multi-adjoint object-oriented concept lattices are derivable from Isbell adjunctions, Kan adjunctions and dual Kan adjunctions between quantaloid-enriched categories, respectively. (Lili Shen) 1 (2) If Q = Q is a unital quantale [26] and ϕ is a fuzzy relation between (crisp) sets X and Y (i.e., ϕ is a map X × Y / / Q), then Mϕ, Kϕ and K † ϕ are concept lattices of the fuzzy context (X, Y, ϕ) of (crisp) sets X and Y [1,13,30].(3) If Q = DQ is the quantaloid of diagonals (cf. [11,25,34]) of a unital quantale Q and ϕ is a fuzzy relation between fuzzy sets X and Y (cf. [9, Definition 2.3]), then Mϕ, Kϕ and K † ϕ are concept lattices of the fuzzy context (X, Y, ϕ) of fuzzy sets X and Y [9,28,31].Since 2006, the theory of multi-adjoint concept lattices was introduced by Medina, Ojeda-Aciego and Ruiz-Calviño [16,19,20,21] as a new machinery of FCA and RST unifying several approaches of fuzzy extensions of concept lattices, and it has been studied in a series of subsequent works (see, e.g., [2,3,4,5,17,18]). As the basic notion of this theory, an adjoint triple [19,20,21] (&, ւ, տ) with respect to posets L 1 , L 2 , P satisfies

Isbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions

“…-Mϕ is the multi-adjoint concept lattice of the context (A, B, ϕ) [13,9], -Kψ is the multi-adjoint property-oriented concept lattice of the context (A, B, ψ) [12,9], -K † ζ is the multi-adjoint object-oriented concept lattice of the context (A, B, ζ ) [12,9].…”