NOTES ON MODULAR ORDERED SETS

Abstract

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 ∈ P with b ≤ 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.

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