Outer and inner medians in some small lattices

TL;DR

This paper investigates the properties of median functions in small finite lattices, focusing on their structure, classification, and the distinction between inner and outer medians, with explicit results for lattices up to six elements.

Contribution

It characterizes the outer and inner median lattices for all lattices with up to six elements, revealing their structural relationships and algebraic properties.

Findings

Inner median lattices are closely related to the symmetric part of the equational basis.

Explicit classifications of median lattices for all lattices up to six elements.

Inner medians are term functions, influencing the lattice's algebraic structure.

Abstract

By median we mean a scheme that inputs three element of a lattice, and outputs an element that is an average of the three inputs in a certain sense. The medians of a given finite lattice form a new lattice that is usually larger than the original, but generates a (not necessarily strictly) smaller variety. A median is called inner if it is a term function. The inner median lattice is closely related to the symmetric part of the equational basis of the lattice. We determine the outer and inner median lattices of all lattices of at most six elements.