---
tag: Y203PPTS61
kind: concept
name: Mal'cev term
msc: [08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence-permutable_algebra
---

## Summary

A Mal'cev term of a [[variety|AJNZJD8442]] is a ternary term $p$ such that $p(x, y, y) = x$ and $p(x, x, y) = y$ hold in the variety.
By Mal'cev's theorem, a variety has a Mal'cev term exactly when it is a [[congruence-permutable variety|1B3N9T68YV]].
Groups, rings, modules and quasigroups all have Mal'cev terms.

## Explanations

### L1 human

In a group, the formula $x y^{-1} z$ cancels the middle input against either neighbour.
If the last two inputs are equal, it returns the first, and if the first two are equal, it returns the last.
A Mal'cev term is any formula with this cancelling behaviour.
Its existence makes congruences, the ways of identifying elements, combine in either order with the same result.