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On Planarity of Join-between Lattices
Marijana Gorjanac Ranitović and Andreja Tepavčević

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.

Keywords: Lattice, planar lattice, order, conjugate order, betweenness relation, join-betweenness relation

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