NOTES ON MODULAR ORDERED SETS
Seon Ho Shin · Journal of the Chungcheng Mathematical Society · 2012
Generalizing modular lattices, a concept of modular ordered sets was introduced by Chajda and Rachunek. In this paper, we characterize modular ordered sets as those partially ordered set P satisfying that for $a,\;b,\;c\;{\in}\;P\;with\;b\;{\leq}\;c,\;l(a,\;b)\;=\;l(a,\;c)\;and\;u(a,\;b)\;=\;u(a,\;c)$ imply $b\;=\;c$ . Using this, we obtain a sufficient condition for them. We also discuss the modularity of the Dedekind-MacNeille completions of ordered sets.