On modular lattices of order dimension two

Abstract

In this note, it is shown that a modular lattice has order dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-over-equals 2"> <mml:semantics> <mml:mrow> <mml:mo> ≦ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leqq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if it contains no subset isomorphic to one of five described partially ordered sets.