On Planarity of Join-between Lattices

Abstract

For elements x, y, and z of a lattice, we say that y is join-between x and z if y is the meet of the join of x and y and that of z and y and, in addition, y is less than or equal to the join of x and z. We prove that if each three element anti-chain of a finite lattice L has an element that is join-between the other two elements of this anti-chain, then L is planar.