---
tag: 9879ZMW5AA
kind: concept
name: Majority term
msc: [08B05, 08B10]
specializes: [4CRH8NNFEF]
sources:
  - url: https://en.wikipedia.org/wiki/Majority_function
---

## Summary

A majority term is a ternary term $m(x, y, z)$ that returns the value shared by at least two of its three arguments.
It is the [[near-unanimity term|4CRH8NNFEF]] of the smallest possible arity.
Lattices and Boolean algebras have a majority term, and every variety with a majority term is congruence-distributive.
For a finite algebra with a majority term, the term operations are exactly the operations that preserve the subalgebras of its square.

## Explanations

### L1 human

Three people vote, and the majority term reports the opinion of at least two of them.
On the set $\set{0, 1}$ the majority function is $m(x, y, z) = 1$ exactly when at least two of $x, y, z$ equal $1$.
For example $m(1, 0, 1) = 1$ and $m(0, 0, 1) = 0$.
In a lattice the same role is played by $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, the median of three elements in a chain.