---
tag: K0AWJK6NZH
kind: concept
name: Product order
msc: [06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Product_order
same_as:
  wikidata: Q7247786
  wikipedia: Product order
---

## Summary

The product order on a [[Cartesian product|09CEDJYBKP]] of partially ordered sets compares tuples coordinate by coordinate.
One tuple is below another when each of its coordinates is below the matching coordinate.
It is the product in the category of posets, and products of chains are the model spaces for [[order dimension|JC4J24MF93]].

## Explanations

### L1 human

Compare two students by their marks in two subjects.
Say that the first is behind the second when the first scored at most as well in both subjects.
A student with marks $(70, 80)$ is behind one with $(75, 90)$, but $(70, 90)$ and $(75, 80)$ cannot be compared.
So the product of two total orders is usually not a total order.