Formal Concept Analysis (FCA) transforms a context bigraph, having vertices of type object and attribute, into a lattice digraph, whose vertices and arcs represent formal concepts and their covering relation. The computational complexity of most FCA algorithms is a polynomial function of the numbers of vertices in both the context bigraph and lattice digraph. While the latter quantity is fixed, the former can be decreased by context standardisation, a process which we show is facilitated by the efficient partition refinement algorithm of Spinrad. The Carve algorithm recursively partitions the context bigraph and corresponding lattice digraph by removing universal objects and attributes and their orphans and partitioning the resultant sub-context into its connected components. The associated software prototype uses the resultant tree structure to support coordinated browsing of both. This paper describes an additional, coordinated representation of the context bigraph which makes explicit in its bi-adjacency matrix the pattern of nested sub-contexts discovered by the Carve algorithm. We show that permuting this matrix into doubly-lexical order with the aid of Spinrad’s algorithm groups together into nested rectangles the bigraph edges belonging to these sub-contexts, and facilitates the two key processing steps of the Carve algorithm.
We use a formal concept theoretic approach to computationally validate the classification of all 192 centralising monoids on three-element sets by Machida and Rosenberg. We determine a manageable finite (and row-reduced) context the intents of which are exactly all centralising monoids on {0,1,2} . As an advantage of our method, we are able to compute a list with a witness for each of the 192 monoids, which has not been available to date and allows for an immediate verification of the figure 192 as a lower bound for the number of centralising monoids. We also confirm that the 192 centralising monoids split into 48 conjugacy classes and we provide a list of all conjugacy types including witnesses.
In previous works we proved that fuzzy closure structures such as closure systems, fuzzy closure systems, fuzzy closure operators and fuzzy closure relations are formal concepts of five Galois connections (three antitone and two isotone), together with the commutativity of the fuzzy part of the diagram. In this work we study the commutativity of the crisp diagram formed by three of these Galois connections.
Implications in Formal Concept Analysis (FCA), Horn clauses in Logic, and Functional Dependencies (FDs) in the Relational Database Model, are very important dependency types in their respective fields. Moreover, they have been proved to be equivalent from a syntactical point of view. Then notions and algorithms related to one dependency type in a field can be reused and applied to another dependency type in the other field. One of these notions is that of cover, also known as a basis, i.e., a compact representation of a complete set of implications, FDs, or Horn clauses. Although the notion of cover exists in the three fields, the characterization and the related uses of a cover are different. In this paper, we study and compare, from an FCA perspective, the principles on which rely the most important covers in each field. Finally, we discuss some open questions that are of interest in the three fields, and especially to the FCA community.
The Hasse diagrams of Formal Concept Analysis (FCA) concept lattices have the disadvantages that users need to be trained in reading the diagrams and diagrams of larger lattices tend to be too cluttered to be comprehensible. This paper therefore discusses how to reduce lattices and then represent them with a specific type of Euler diagram instead of Hasse diagrams. A semi-automated process of reducing concept lattices is described and supported by algorithms.
