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Discussiones Mathematicae General Algebra and Applications 22(2) (2002) 167-181
doi: 10.7151/dmgaa.1056

CLASSIFICATION SYSTEMS AND THEIR LATTICE

Sándor Radeleczki^∗

Institute of Mathematics, University of Miskolc
3515 Miskolc-Egyetemváros, Hungary
e-mail: matradi@gold.uni-miskolc.hu

Abstract

We define and study classification systems in an arbitrary CJ-generated complete lattice L. Introducing a partial order among the classification systems of L, we obtain a complete lattice denoted by Cls(L). By using the elements of the classification systems, another lattice is also constructed: the box lattice B(L) of L. We show that B(L) is an atomistic complete lattice, moreover Cls(L)=Cls(B(L)). If B(L) is a pseudocomplemented lattice, then every classification system of L is independent and Cls(L) is a partition lattice.

Keywords: concept lattice, CJ-generated complete lattice, atomistic complete lattice, (independent) classification system, classification lattice, box lattice.

2000 AMS Mathematics Subject Classification: Primary 06B05, 06B15; Secondary 06B23.

References

[1]

R. Beazer, Pseudocomplemented algebras with Boolean congruence lattices, J. Austral. Math. Soc. 26 (1978), 163-168.

[2]

B. Ganter and R. Wille, Formal Concept Analysis. Mathematical Foundations, Springer-Verlag, Berlin 1999.

[3]

S. Radeleczki, Concept lattices and their application in Group Technology (Hungarian), p. 3-8 in: ``Proceedings of International Computer Science Conference: microCAD'98 (Miskolc, 1998)'', University of Miskolc 1999.

[4]

S. Radeleczki, Classification systems and the decomposition of a lattice into direct products, Math. Notes (Miskolc) 1 (2000), 145-156.

[5]

E.T. Schmidt, A Survey on Congruence Lattice Representations, Teubner-texte zur Math., Band 42, Leipzig 1982.

[6]

M. Stern, Semimodular Lattices. Theory and Applications, Cambridge University Press, Cambridge 1999.

[7]

R. Wille, Subdirect decomposition of concept lattices, Algebra Universalis 17 (1983), 275-287.

Received 3 October 2002