Three generalisations of lattice distributivity: an FCA perspective

In the present study we investigate concept lattices and their corrsponding formal contexts K =(G, M, I) that are usually given by a cross-table and from wich we infer a set of implications. The triple K consists of two sets, namely the set of objects G, which refers to the rows of the table, and the set of attributes M, which refers to the names of the columns of the crosstable. The triple is complete with an incidence relation I that states that an object g possesses an attribute m ifandonlyifinrowgthere is a cross in the column m. The concept lattice B(K) arises from K =(G, M, I) by collecting all maximal rectangles, which are full of crosses (regardless of succession of rows and columns). Such a maximal rectangle is characterised by the collection of its objects A ⊆ G and its attributes B ⊆ M. The pair (A, B) is called formal concept. All formal concepts can naturally be ordered and thereby form a complete lattice.