Extent partitions and context extensions

Abstract

ABSTRACT. We prove that the extent partitions of a formal context K := (G, M, I) can be constructed from the box extents of it, which form a complete atomistic lattice. K is called a one-object extension of the subcontext (H, M, J) if it is obtained by adding a new element with attributes in M to the set H. We investigate the interplay between the box extents of (H, M, J) and those of its one-object extension K, and describe those extent partitions of (H, M, J) which can be extended to K. Extent partitions and classification systemsA partition of a set G is a family of nonempty, mutually disjoint subsets of G whose union is G, i.e. a familywhere T is some index set) withThe sets G t are called the classes of the partition π.2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 06B23; Secondary 0B05, 06B15. K e y w o r d s: concept lattice, extent partition, box extent, one-object extension. The research of the second and third author was carried out as part of the TAMOP-4.2.1. B-10/2/KONV., and respectively TAMOP 4.2

Classification trees in a box extent lattice

“…We say that K = (G, M, I) is a one-object extension of K H = (H, M, I ∩ H × M ), where H ⊆ G, if there exists a z ∈ G such that H = G \ {z} . In this section we use the results of [3], where the authors studied changes of the box extents of a one-object extension of the context. In [3], it was shown that the intersection of box extents is also a box extent.…”

“…In this section we use the results of [3], where the authors studied changes of the box extents of a one-object extension of the context. In [3], it was shown that the intersection of box extents is also a box extent. In [3], it was also proved that any extent partition of the subcontext is also an extent partition of K and the box extents of the subcontext are also box extents of K and the following propositions:…”

“…The set of all box extents of L is denoted by B(G, M, I). The box extents are ordered by inclusion and in [3] was proved that B(G, M, I) is a complete atomistic lattice which is a ∧-subsemilattice of the concept lattice. (Note that if ∅ = ∅, then {G} is the only extent partition of the context (G, M, I)).…”

“…In [3] the box extents of a context (G, M, I) where defined as the sets E belonging to some extent partition of (G, M, I) or E = ∅ . The set of all box extents of L is denoted by B(G, M, I).…”

Classification lattices are geometric for complete atomistic lattices

“…Though this problem is partly solved by Mao [5], it is unsolved completely to date. This problem is related to the application of concept lattices to one of the main problems in group technology (see [2,[6][7][8]). Actually, the problem is also related with the study of CD-bases of a lattice (see [9,10]), or to the investigation of the decomposition systems of a closure system (cf.…”