PDF

Discussiones Mathematicae General Algebra and Applications 20(1) (2000) 43-49
DOI: https://doi.org/10.7151/dmgaa.1004

ON DUALITY OF SUBMODULE LATTICES

Gábor Czédli and Géza Takách

JATE Bolyai Institute
Aradi vértanúk tere 1, H-6720 Szeged, Hungary
e-mail: czedli@math.u-szeged.hu
e-mail: takach@math.u-szeged.hu

Dedicated to the memory of George Hutchinson

Abstract

An elementary proof is given for Hutchinson's duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.

Keywords: submodule lattice, lattice identity, duality.

1991 Mathematics Subject Classification: Primary 06C05, Secondary 08B10, 16D99.

References

[1]	G. Frobenius, Theorie der linearen Formen mit ganzen Coefficienten, J. Reine Angew. Math. 86 (1879), 146-208.
[2]	G. Hutchinson, On classes of lattices representable by modules, p. 69-94 in: Proceedings of the University of Houston Lattice Theory Conference, Univ. Houston 1973.
[3]	G. Hutchinson and G. Czédli, A test for identities satisfied in submodule lattices, Algebra Universalis 8 (1978), 269-309.
[4]	A.F. Pixley, Local Mal'cev conditions, Canadian Math. Bull. 15 (1972),559-568.
[5]	R. Wille, Kongruenzklassengeometrien, Lecture Notes in Math., no. 113, Springer-Verlag, Berlin-Heidelberg-New York 1970.

Received 23 February 1998