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The descending chain condition in modular lattices

Published online by Cambridge University Press:  09 April 2009

Thomas G. Newman
Thomas G. Newman
Affiliation:
Texas Tech UniversityLubbock, Texas, 19409, U.S.A.
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In a recent paper Kovács [1] studied join-continuous modular lattices which satisfy the following conditions: (i) every element is a join of finitely many join-irredicibles, and, (ii) the set of join-irreducibles satisfies the descending chain condition. He was able to prove that such a lattice must itself satisfy the descending chain condition. Interest was expressed in whether or not one could obtain the same result without the assumption of modularity and/or of join-continuity. In this paper we give an elementary proof of this result without the assumption of join- continuity (which of course must then follow as a consequence of the descending chain condition). In addition we give a suitable example to show that modularity may not be omitted in general.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 4 , December 1972 , pp. 443 - 444
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Kovács, L. G., ‘The descending chain condition in join continuous modular lattices’ J. Aust. Math. Soc. 10 (1969), 1–4.CrossRefGoogle Scholar