Spectral Lattices of $\mathbb{\overline R}_{\rm max,+}$-Formal Contexts

Abstract

In [13] a generalisation of Formal Concept Analysis was introduced with data mining applications in mind, $\mathcal K$-Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the standard Boolean carrier set. Subsequently, the structural lattice of such generalised contexts was introduced in [15], to provide a limited equivalent to the main theorem of $\mathcal K$-Formal Concept Analysis, resting on a crucial parameter, the degree of existence of the object-attribute pairs ϕ. In this paper we introduce the spectral lattice of a concrete instance of $\mathcal K$-Formal Concept Analysis, as a further means to clarify the structural and the $\mathcal K$-Concept Lattices and the choice of ϕ. Specifically, we develop techniques to obtain the join- and meet-irreducibles of a $\mathbb{\overline R}_{\rm max,+}$-Concept Lattice independently of ϕ and try to clarify its relation to the corresponding structural lattice.