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

Abstract

Recently, Denniston, Melton, and Rodabaugh presented a new categorical outlook on a certain lattice-valued extension of Formal Concept Analysis (FCA) of Ganter and Wille; their outlook was based on the notion of lattice-valued interchange system and a category of Galois connections. This paper extends the approach of Denniston et al. clarifying the relationships between Chu spaces of Pratt, many-valued formal contexts of FCA, lattice-valued interchange systems, and Galois connections.

Operators on Pavelka's algebras induced by fuzzy relations

“…In the following theorem we characterize φ, ρ and δ-type mappings as mappings induced by fuzzy relations. A result similar to the second part has been proved for quantals by S. Solovjovs [29].…”

Fuzzy quantic nuclei and conuclei with applications to fuzzy semi-quantales and (L, M)-quasi-fuzzy topologies

“…For the convenience of the reader, the following proposition recalls some of their other properties, which will be heavily used throughout this paper. Proposition 1 [13,19] For L ∈ |Quant| with a, b, c ∈ L and B ⊆ L, we have the following properties: Before going too much further, we recall that if L = (L, ≤) is a poset, an order preserving function g : L → L is called a closure (resp., coclosure) operator on the poset L = (L, ≤) [13,17] iff it satisfies the following conditions: [20] Let L be a quantale. A non-empty subset I ⊆ L is called a left (resp., right) ideal of L if it satisfies the following three conditions:…”

Lattice-valued bornological systems

“…If the tissue has a tumor, then one can approximate this tumor with a family of sets of slices {Z i ⊆ H | i ∈ I }, calculate the Hausdorff dimension of the elements of this family, and obtain thereby an estimation (more precisely, the statistical inference, which is studied in [33] and the references therein) of its (tumor) level of carcinogenicity (see the last paragraph of Section 1). To be more precise, we elaborate this topic (in a somewhat simplified manner) in the next example, in which we follow the notation of Example 7.…”

“…Lattice-valued analogues of the main system-related procedures, i.e., spatialization (a space from a system) and localification (a locale from a system) were soon considered in [28,29,31], thereby providing a complete fuzzification of the original setting of S. Vickers. At present, the theory of lattice-valued topological systems has already found applications in several scientific fields [7,8,11,12,30,32], including, e.g., formal concept analysis, topology, theoretical computer science, and theoretical physics. With Remark 2 in mind, we introduce the category of the underlying algebraic structures of bornologies.…”