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.
Highlights
This paper extends the approach of Denniston et al clarifying the relationships between Chu spaces of Pratt, many-valued formal contexts of Formal Concept Analysis (FCA), lattice-valued interchange systems, and Galois connections
This paper considers a particular application of the theory of variety-based topological systems, introduced in [1] as a generalization of topological systems of Vickers [2], which in turn provide a common framework for both topological spaces and their underlying algebraic structures frames, thereby allowing researchers to switch freely between the spatial and localic viewpoints
Parallel to the previously mentioned developments, we introduced the concept of variety-based topological system [1], which extended the setting of Denniston et al, and included the case of state property systems of Aerts [16,17,18], considered in [19] in full detail, and we brought the functor of Journal of Mathematics
Summary
This paper considers a particular application of the theory of variety-based topological systems, introduced in [1] as a generalization of topological systems of Vickers [2], which in turn provide a common framework for both topological spaces and their underlying algebraic structures frames ( called locales), thereby allowing researchers to switch freely between the spatial and localic viewpoints. The authors brought their theory into maturity in [9], where they considered its applications to both lattice-valued variable-basis topology of Rodabaugh [4] and (L, M)-fuzzy topology of Kubiak and Sostak [10] Later on, they introduced a particular instance of their concept called interchange system [11], which was motivated by certain aspects of program semantics (the socalled predicate transformers) initiated by Dijkstra [12]. It never gives much attention to the links between interchange systems and Chu spaces, thereby making no use of the well-developed machinery of the latter While pursuing their approach to many-valued FCA, the authors never mention the already existing many-valued formal contexts of Ganter and Wille [31], which (being similar to Chu spaces) essentially extend their own. We tried to make the paper as much self-contained as possible, it is expected from the reader to be acquainted with basic concepts of category theory, that is, those of category and functor
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