Geodesic
Chain conditions on posets
Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 79-81

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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. 
Geoffroy, Cynthia D.; Scheiblich, H. E. Chain conditions on posets. Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 79-81. doi: 10.1017/S0017089500002172
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[1] 1.Birkhoff, Garrett, Lattice Theory, Amer. Math. Soc. Colloquium Publications 25 (New York, 1948). Google Scholar

[2] 2.Rhodes, Joe B., Modular and Distributive Semilattices, Ph.D. Thesis, University of Texas. Google Scholar

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