---
tag: 6068BQ4XZF
kind: definition
name: Absorbing element
msc: [08A02, 20M99]
defines: PJ61DXYG2C
uses: [0M418FE7BB]
sources:
  - url: https://en.wikipedia.org/wiki/Absorbing_element
---

## Definition

Let $\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$ and let $z \in S$.

1. The element $z$ is *left absorbing* if $z \ast a = z$ for all $a \in S$.
2. The element $z$ is *right absorbing* if $a \ast z = z$ for all $a \in S$.
3. The element $z$ is *absorbing*, or a *zero element*, if it is both left and right absorbing.

## Remarks

Example: in $(\Z, \cdot)$ the integer $0$ is absorbing, because $0 \cdot a = a \cdot 0 = 0$ for all $a \in \Z$.

Non-example: in $(\Z, +)$ the integer $0$ is not absorbing, because $0 + 1 = 1 \ne 0$; it is an identity element instead.

If $z$ is left absorbing and $z'$ is right absorbing, then $z = z \ast z' = z'$.
So an operation has at most one absorbing element.
On a set with at least two elements, the operation $a \ast b = a$ makes every element left absorbing and no element right absorbing.