Planar semilattices and nearlattices with eighty-three subnearlattices

TL;DR

This paper establishes a condition linking the number of subnearlattices in finite nearlattices to their planarity, providing a sharp threshold for when such structures have planar Hasse diagrams.

Contribution

It proves a sharp bound connecting the number of subnearlattices to the planarity of finite nearlattices, enriching the understanding of their structural properties.

Findings

Nearlattices with many subnearlattices are planar.

The bound of 83·2^{n-8} subnearlattices is sharp for n > 8.

Finite nearlattices are equivalent to several other algebraic structures.

Abstract

Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an $n$-element nearlattice has at least $83\cdot 2^{n-8}$ subnearlattices, then it has a planar Hasse diagram. For $n>8$, this result is sharp.