Geodesic
General existence conditions for maximal elements of ordered sets
Trudy Instituta matematiki, Tome 15 (2007) no. 2, pp. 3-14

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It is proved that Arkhangelski principle of existence of top-ascending well-ordered chain is equivalent to the statement of Gajek–Zagrodny lemma. Both the criterium of nonemptyness and the criterium of cofinality for the subset of maximal elements of an ordered set are derived on the base of Arkhangelski principle. 
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E. V. Gireiko; V. V. Gorokhovik. General existence conditions for maximal elements of ordered sets. Trudy Instituta matematiki, Tome 15 (2007) no. 2, pp. 3-14. http://geodesic.mathdoc.fr/item/TIMB_2007_15_2_a0/