Classification systems
by
Sándor Radeleczki
Institute of Mathematics, University of Miskolc, Hungary

Given a row-reduced context (G, M, I) a classification system is a set of elements {a_i}_{i in I} subset L(G, M, I) such that x = \/ _{i in I}(x /\ a_i), for all x in L and such that a_i /\ a_j=0_{\UNICODE[m]0xa3}, whenever i =/= j. Making abstraction from the original problem we define and study classification system in an arbitrary CJ-generated complete lattice L. In this case introducing a natural order between the classification systems of L, we obtain a complete semimodular lattice denoted by Cls(L). The elements a_i of L which belong to some classification system {a_i}_{i in I} of L we call the box elements of L. The set of box elements of L with the partial order induced by L is a relative sublattice of L denoted by B(L). We show thatB(L) is a complete atomistic lattice and Cls(L) =~ Cls(B(L)). We prove that Cls(L) is isomorphic to a partition lattice iff B(L) is a Boolean lattice.

Date received: October 27, 2000