The concept lattice functors

Abstract

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.

Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis

“…Additionally, both (many-valued) formal contexts and their morphisms are treated in an algebraic way in [31] (from where our notation for context morphisms, which is different from the system setting, is partly borrowed). There does exist some research on certain categories of contexts (e.g., [57][58][59][60][61]), which, however, does not study the categorical foundation for FCA, but rather concentrates on a fixed categorical framework and its related results. It is the main purpose of this paper, to investigate such possible categorytheoretic foundations and their relationships to each other (leaving their related results to the further study on the topic).…”

Multi-adjoint fuzzy rough sets: Definition, properties and attribute selection

“…We can extend the discernibility function to a mapping from P(A) to L interpreting the infimum in Eq. (25) by an aggregation operator @, replacing the exact equalities by the respective approximate equalities (fuzzy indiscernibility relations), and writing the expression…”

“…On the other hand, formal concept analysis, introduced by Wille in the decade of 1980 [28], arose as another mathematical theory for qualitative data analysis and, currently, has become an interesting research topic both with regard to its mathematical foundations [16,25] and with regard to its multiple applications [5,6].…”

“…d increases. Therefore, f @ (B), expresses the degree to which, for all object pairs, different values in attributes of B correspond to different values of d.The previous definition of discernibility function is a fuzzy generalization of the crisp one(25) and the fuzzy ones introduced in [3], assuming consistency.…”

Multi-adjoint concept lattices via quantaloid-enriched categories

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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.

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