Every relation between posets gives rise to an adjunction, known as a Galois connection, between the corresponding power sets. Formal concept analysis (FCA) studies the fixed points of these adjunctions, which can be interpreted as latent
“concepts” [20], [19]. In [47] Pavlovic defines a generalisation of posets he calls proximity sets (or proxets), which are equivalent to the generalised metric spaces of Lawvere [37], and introduces a form of quantitative concept analysis (QCA) which provides a different viewpoint from other approaches to fuzzy concept analysis (for a survey see [4]).
The nucleus of a fuzzy relation between proxets is defined in terms of the fixed points of a naturally arising adjunction based on the given relation, generalising the Galois connections of formal concept analysis. By giving the unit interval [0, 1] an appropriate category structure it can be shown that proxets are simply [0, 1]-enriched categories and the nuclues of a proximity relation between proxets is a generalisation of the notion of the Isbell completion of an enriched category.
We prove that the sets of fixed points of an adjunction arising from a fuzzy relation can be given the structure of complete idempotent semimodules and show that they are isomorphic to tropical convex hulls of point configurations in tropical projective space, in which addition and scalar multiplication are replaced with pointwise minima and addition, respectively. We show that some the results of Develin and Sturmfels on tropical convex sets [13] can be applied to give the nucleus of a proximity relation the structure of a cell complex, which we term the fuzzy concept complex. We provide a formula for counting cells of a given dimension in generic situations.
We conclude with some thoughts on computing the fuzzy concept complex using ideas from Ardila and Develin’s work on tropical oriented matroids [1].
