The Linear Algebra in Extended Formal Concept Analysis Over Idempotent Semifields

Abstract

We report on progress relating $\mathcal K$-valued FCA to $\mathcal K$-Linear Algebra where $\mathcal K$ is an idempotent semifield. We first find that the standard machinery of linear algebra points to Galois adjunctions as the preferred construction, which generates either Neighbourhood Lattices of attributes or objects. For the Neighbourhood of objects we provide the adjoints, their respective closure and interior operators and the general structure of the lattices, both of objects and attributes. Next, these results and those previous on Galois connections are set against the backdrop of Extended Formal Concept Analysis. Our results show that for a $\mathcal K$-valued formal context (G, M, R)—where $|G| = g$, $|M| = m$ and $R \in K^{g\times {m}}$—there are only two different “shapes” of lattices each of which comes in four different “colours”, suggesting a notion of a 4-concept associated to a formal concept. Finally, we draw some conclusions as to the use of these as data exploration constructs, allowing many different “readings” on the contextualized data.

C. Peláez-Moreno—FVA and CPM have been partially supported by the Spanish Government-MinECo projects TEC2014-53390-P and TEC2014-61729-EXP for this work.