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Chain conditions on posets

Published online by Cambridge University Press:  18 May 2009

Cynthia D. Geoffroy  and
H. E. Scheiblich
Cynthia D. Geoffroy
Affiliation:
University of South Carolina, Columbia, South Carolina
H. E. Scheiblich
Affiliation:
University of South Carolina, Columbia, South Carolina
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The aim of this note is to generalize to an arbitrary partially ordered set (poset) (P, ≦) the standard lattice results on the Jordan–Dedekind Chain Condition (abbreviated hereafter to J.D.C.C.). Birkhoff [1] defines semimodularity for a lattice L by

(ξ) if x, y cover a and x # y, then x ∨ y covers x and y.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 79 - 81
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Birkhoff, Garrett, Lattice Theory, Amer. Math. Soc. Colloquium Publications 25 (New York, 1948).Google Scholar
2.Rhodes, Joe B., Modular and Distributive Semilattices, Ph.D. Thesis, University of Texas.Google Scholar