---
tag: SHMTFRH28E
kind: concept
name: Gumm terms
msc: [08B05, 08B10]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_lattice
---

## Summary

Gumm terms are a finite chain of ternary terms of a [[variety|AJNZJD8442]].
They combine a chain like [[Jónsson terms|3M8Y76E4AG]] with a final step like a [[Mal'cev term|Y203PPTS61]].
A variety has Gumm terms if and only if it is a [[congruence-modular variety|QAQAS3JWXW]].
This form of the Mal'cev condition for modularity shows that congruence modularity mixes the two basic cases, congruence distributivity and congruence permutability.

## Explanations

### L1 human

Lattices have a chain of formulas of one kind, and groups have one formula of another kind.
Gumm terms allow a chain of the lattice kind followed by one formula of the group kind.
A variety has such a chain exactly when all its congruence lattices satisfy the modular law.
So congruence modularity sits between the lattice-like and the group-like behaviour, and contains both.