---
tag: PJ61DXYG2C
kind: concept
name: Absorbing element
msc: [08A02, 20M99]
sources:
  - url: https://en.wikipedia.org/wiki/Absorbing_element
same_as:
  wikidata: Q187420
  wikipedia: Absorbing element
---

## Summary

An absorbing element, or zero element, for a [[binary operation|0M418FE7BB]] is an element that turns every product it enters into itself.
The number $0$ is absorbing for multiplication of numbers, and the empty set is absorbing for intersection.
A binary operation has at most one two-sided absorbing element.
Absorbing elements are central in semigroup theory and in rings, where $0$ is absorbing for multiplication.

## Explanations

### L1 human

An absorbing element swallows everything it meets.
Multiply any number by $0$ and you get $0$.
Intersect any set with the empty set and you get the empty set.
For the operation $\max$ on the interval $[0, 1]$, the number $1$ is absorbing, since $\max(x, 1) = 1$.