---
tag: 6AJ7QQCTQQ
kind: concept
name: Ordinal sum of posets
msc: [06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Partially_ordered_set
---

## Summary

The ordinal sum of two partially ordered sets places the second one entirely above the first.
Inside each part the order is unchanged, and every element of the first part is below every element of the second.
It is also called the linear sum or series composition, and it extends the addition of [[ordinal numbers|0YSCSX5YZC]].

## Explanations

### L1 human

Stack two posets on top of each other, like two floors of a building.
Everything on the lower floor is below everything on the upper floor.
Stacking a single point on top of a two-element antichain gives a "V" shape turned upside down, with the point as the top.
The order of stacking matters: a point below the antichain gives a different poset.