---
tag: B0YKA8401M
kind: concept
name: Day terms
msc: [08B05, 08B10]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_lattice
---

## Summary

Day terms are a finite chain of quaternary terms of a [[variety|AJNZJD8442]] that satisfy certain identities.
By Day's theorem, a variety has Day terms if and only if it is a [[congruence-modular variety|QAQAS3JWXW]].
They give a Mal'cev condition for congruence modularity, in the same way that [[Jónsson terms|3M8Y76E4AG]] do for congruence distributivity.

## Explanations

### L1 human

Day terms are a list of formulas with four inputs that step from the first input to the last one.
Each formula returns $x$ when its inputs have the pattern $x, y, y, x$.
Neighbouring formulas agree on inputs of certain patterns with repeated values.
Their existence is exactly what makes all congruence lattices in the variety modular.