---
tag: 44HTG620GB
kind: definition
name: Latin square
msc: [05B15]
defines: CDVRMVED2V
uses: [0N1MY1T0W9, 0BBTH3YSZA]
sources:
  - url: https://en.wikipedia.org/wiki/Latin_square
---

## Definition

Let $n$ be a positive integer and let $S$ be a [[finite set|0BBTH3YSZA]] with $n$ elements, the symbols.
A Latin square of order $n$ on $S$ is an $n \times n$ array $L = (L_{ij})$ with entries in $S$ such that:

1. In each row, each element of $S$ occurs exactly once.
2. In each column, each element of $S$ occurs exactly once.

## Remarks

Example: for $i, j \in \set{0, 1, \dots, n - 1}$, put $L_{ij} = (i + j) \bmod n$.
In row $i$ the map $j \mapsto i + j \bmod n$ is a bijection of $\set{0, \dots, n-1}$, so each symbol occurs once; the same holds for each column.
This square is the addition table of the [[integers modulo n|01N3AYAA6N]].
More generally, the Cayley table of each finite [[group|G5S1M9]] is a Latin square.
Non-example: the array with rows $1\,2$ and $1\,2$ is not a Latin square, because the symbol $1$ occurs twice in the first column.
A Latin square is the same thing as the multiplication table of a finite quasigroup.
Two Latin squares can be combined into a pair of [[orthogonal Latin squares|H6TN6BADVY]].