Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires *concept mining* and *concept analysis* to extract shared semantics for different network nodes. Powerful semantic analysis systems have been the drivers of nearly all paradigm shifts on the web. In categorical terms, most of them can be described as bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style completions from posets to suitably enriched categories. Yet it has been well known for more than 40 years that ordinary categories themselves in general do not permit such completions. Armed with this new semantical view of Dedekind-MacNeille completions, and of matrix bicompletions, we take another look at this ancient mystery. It turns out that simple categorical versions of the *limit superior* and *limit inferior* operations characterize a general notion of Dedekind-MacNeille completion, that seems to be appropriate for ordinary categories, and boils down to the more familiar enriched versions when the limits inferior and superior coincide. This explains away the apparent gap among the completions of ordinary categories, and broadens the path towards categorical concept mining and analysis, opened in previous work.Comment: 22 pages, 5 figures and 9 diagram

Formal Concept Analysis with Constraints by EM Operators

Formal concept analysis is a method of exploratory data analysisthat aims at the extraction of natural cluster from object-attributedata tables. We present a way to add user's background knowledge toformal concept analysis. The type of background knowledge we dealwith relates to relative importance of attributes in the input data.We introduce EM operators which constrain in attributes of formalconcept analysis. The main aim is to make extraction of conceptsfrom the input data more focused by taking into account thebackground knowledge. Particularly, only concepts which arecompatible with the constraint are extracted from data. Therefore,the number of extracted concepts becomes smaller since we leave outnon-interesting concepts. We concentrate on foundational aspectssuch as mathematical feasibility and computational tractability

Generalization of One-Sided Concept Lattices

We provide a generalization of one-sided (crisp-fuzzy) concept lattices, based on Galois connections. Our approach allows analysis of object-attribute models with different structures for truth values of attributes. Moreover, we prove that this method of creating one-sided concept lattices is the most general one, i.e., with respect to the set of admissible formal contexts, it produces all Galois connections between power sets and the products of complete lattices. Some possible applications of this approach are also included