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Outer and inner medians in some small lattices

Leen Aburub, Gergo Gyenizse

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.

Outer and inner medians in some small lattices

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.
Paper Structure (5 sections, 8 theorems, 16 equations, 2 figures, 1 table)

This paper contains 5 sections, 8 theorems, 16 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and general observations
  3. On the outer median lattice
  4. Outer and inner median lattices of small lattices
  5. Symmetric characterization of modularity

Key Result

Proposition 4

If two lattices generate the same variety, then their inner median lattices are isomorphic.

Figures (2)

  • Figure 1: The lattice $\mathbf{E}_{n}$ with $n$ medians and its T-poset
  • Figure 2: The lattices $\mathbf{L}_4$ and $\mathbf{L}_5$, and the poset $\mathcal{T}_{\mathbf{L}_4}$

Theorems & Definitions (21)

  • Conjecture 1
  • Conjecture 2
  • Definition 3
  • Proposition 4
  • Definition 5
  • Theorem 6
  • proof
  • Definition 7
  • Definition 8
  • Proposition 9
  • ...and 11 more