---
tag: 7SP03R55HW
kind: definition
name: Mal'cev condition
msc: [08B05]
defines: 6R0CA4JWWY
uses: [AJNZJD8442, PKD440G2XW, N9M2N27END]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence-permutable_algebra
---

## Definition

A *strong Mal'cev condition* is given by finitely many operation symbols $g_1, \dots, g_k$ with arities and a finite set $\Sigma$ of [[identities|PKD440G2XW]] in these symbols.
A [[variety|AJNZJD8442]] $\mathcal{V}$ *satisfies* it if $\mathcal{V}$ has terms $t_1, \dots, t_k$ of the same arities with the following property.
Each identity of $\Sigma$ holds in $\mathcal{V}$ after each $g_j$ is replaced by $t_j$.
A *Mal'cev condition* is a sequence $U_1, U_2, \dots$ of strong Mal'cev conditions such that every variety that satisfies $U_n$ also satisfies $U_{n+1}$, for each $n$.
A variety satisfies the Mal'cev condition if it satisfies $U_n$ for some $n$.
A property of varieties is *Mal'cev definable* if some Mal'cev condition is satisfied by exactly the varieties with that property.

## Remarks

Example: the existence of a [[Mal'cev term|Y203PPTS61]] is a strong Mal'cev condition, with one ternary symbol $p$ and $\Sigma = \set{p(x, y, y) = x, \ p(x, x, y) = y}$.
The variety of groups satisfies it with $t(x, y, z) = x y^{-1} z$, since $x y^{-1} y = x$ and $x x^{-1} z = z$.
The existence of [[Jónsson terms|3M8Y76E4AG]] for some $n$ is a Mal'cev condition, with one strong Mal'cev condition $U_n$ for each $n$.
Non-example: local finiteness of a variety is not Mal'cev definable.
The variety of sets, with no operations, is locally finite, and its terms are the projections.
If the projections satisfy $\Sigma$, then the projections of every variety satisfy $\Sigma$, so every variety would satisfy the condition.
But the variety of groups is not locally finite.

## Explanations

### L1 human

A Mal'cev condition says that a variety has some terms that satisfy a fixed list of equations, possibly with the length of the list allowed to grow.