---
tag: YNM6V0R269
kind: definition
name: Gumm terms
msc: [08B05, 08B10]
defines: SHMTFRH28E
uses: [AJNZJD8442, PKD440G2XW, QAQAS3JWXW, Y203PPTS61, 3M8Y76E4AG]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_lattice
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $n \ge 0$.
*Gumm terms* for $\mathcal{V}$ are ternary terms $d_0, d_1, \dots, d_n$ and $p$ such that $\mathcal{V}$ satisfies the following [[identities|PKD440G2XW]]:

1. $d_0(x, y, z) = x$.
2. $d_i(x, y, x) = x$ for every $i \le n$.
3. $d_i(x, x, z) = d_{i+1}(x, x, z)$ for every even $i < n$.
4. $d_i(x, z, z) = d_{i+1}(x, z, z)$ for every odd $i < n$.
5. $d_n(x, z, z) = p(x, z, z)$.
6. $p(x, x, z) = z$.

## Remarks

A variety has Gumm terms if and only if it is a [[congruence-modular variety|QAQAS3JWXW]].
Example: if $p$ is a [[Mal'cev term|Y203PPTS61]], then $n = 0$, $d_0 = x$ and $p$ are Gumm terms, since $p(x, z, z) = x = d_0(x, z, z)$ and $p(x, x, z) = z$.
Example: lattices have Gumm terms with $n = 1$, $d_0 = x$, $d_1 = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ and $p(x, y, z) = z$.
Here $d_0(x, x, z) = x = d_1(x, x, z)$, $d_1(x, z, z) = z = p(x, z, z)$, and $p(x, x, z) = z$.
These examples show the two extreme cases: a pure Mal'cev step, and a chain of [[Jónsson terms|3M8Y76E4AG]] with $p$ the third projection.
Non-example: with $n = 0$, the terms $d_0 = x$ and $p = x$ are not Gumm terms in a nontrivial variety, since clause 6 reads $x = z$.

## Explanations

### L1 human

Gumm terms are a chain of three-input formulas, ending in one Mal'cev-like formula, that witness modularity of all congruence lattices by equations alone.