Distributive lattices, associative geometries: the arithmetic case

TL;DR

This paper establishes a new identity in distributive lattices, especially in the natural numbers with gcd and lcm, revealing many semigroup structures that resemble non-commutative analogs of modular rings.

Contribution

It introduces a novel identity characterizing distributive lattices and explores their semigroup structures, including explicit multiplication tables in the arithmetic case.

Findings

Identifies a key identity valid in natural numbers with gcd and lcm.

Shows that fixing three arguments yields associative products in distributive lattices.

Finds many semigroups are periodic and resemble non-commutative rings.

Abstract

We prove an identity for five arguments, valid in the lattice of natural numbers with gcd and lcm as lattice operations. More generally, this identity characterizes arbitrary distributive lattices. Fixing three of the five arguments, we always get associative products, and thus every distributive lattice carries many semigroup structures. In the arithmetic case, we explicitly compute multiplication tables of such semigroups and describe some of their properties. Many of them are periodic, and can be seen as "non-commutative analogs" of the rings Z/nZ.