Geodesic
The conditions for the convergence of isotonic operators in partially ordered sets with convergence classes
Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 337-348 Cet article a éte moissonné depuis la source Math-Net.Ru

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The necessary and sufficient conditions are given which must be satisfied by an element $x$ of a partially ordered set $X$ with convergence close to topological in order that every net of isotonic operators of the set $X'\subset X$ in $X$, converging on some set $G\subset X'$ to the identity operator, should converge on $X$ to $x$. 
@article{MZM_1972_12_3_a14,
     author = {R. K. Vasil'ev},
     title = {The conditions for the convergence of isotonic operators in partially ordered sets with convergence classes},
     journal = {Matemati\v{c}eskie zametki},
     pages = {337--348},
     year = {1972},
     volume = {12},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_3_a14/}
}
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R. K. Vasil'ev. The conditions for the convergence of isotonic operators in partially ordered sets with convergence classes. Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 337-348. http://geodesic.mathdoc.fr/item/MZM_1972_12_3_a14/