Abstract
Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯-four-fold Formal Concept Analysis (K¯-4FCA) where K¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of K¯-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results.
Highlights
This paper tries to show how Linear Algebra over idempotent semifields in general— and Galois connections in particular—contributes to the program of Lattice Computing (LC) [1] and its attempt to provide “an evolving collection of tools and methodologies that process lattice-ordered data”, as a means of establishing an information processing paradigm belonging to the wider field of Computational Intelligence [2], with an explicit aim at modelling Cyber–Physical Systems [3]
Note that this paper has a sibling paper that discusses the Fundamental Theorem of Linear Algebra over idempotent semifields and its relation to an idempotent Singular Value Decomposition and matrix reconstruction [6], but in this paper we restrict ourselves to the issues that are pertinent to all flavours of K-Formal Concept Analysis (FCA) and their duals K-Formal Concept Analysis (K-FCA)
We investigate in it: (1) the four-fold Galois connections related to an idempotent semifield-valued matrix; (2) what are the implications of a change of bias, that is, using K for the analysis; (3) the relation to LC, in general, and to Formal Concept Analysis, in particular, and (4) the relationship of these techniques with other types of FCA
Summary
This paper tries to show how Linear Algebra over idempotent semifields in general— and Galois connections in particular—contributes to the program of Lattice Computing (LC) [1] and its attempt to provide “an evolving collection of tools and methodologies that process lattice-ordered data”, as a means of establishing an information processing paradigm belonging to the wider field of Computational Intelligence [2], with an explicit aim at modelling Cyber–Physical Systems [3]. At the same time—as this paper will show—Linear Algebra constructs over fields produce analogues over idempotent semifields that give rise to (complete) lattices, so we almost always end up “computing with lattices” This idea is strengthened by the fact that the ternary semiring in Example 1 that subsumes the boolean semiring is embedded in every complete idempotent semifield, enabling a direct generalization of boolean constructions from Discrete Algebra—sets, graphs, formal power series, etc. The top completion provides not one but a pair of dually-ordered semifields (K, K) with the inversion in the multiplicative group acting as the duality inducing operator. This entails that expressions will have two (differently completed) multiplications, additions, etc., causing a notational problem that has not yet been agreed upon in the community (see the Appendix A in [6]). This, as well, has historical reasons that have led some researchers to postulate Linear Algebra over the max-plus semifield as the “algebra of combinatorics” [12]
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