You are here
Homeabsorbing element
Primary tabs
absorbing element
An element $\zeta$ of a groupoid $(G,\,*)$ is called an absorbing element (in French un élément absorbant) for the operation “$*$”, if it satisfies
$\zeta\!*\!a\;=\;a\!*\!\zeta\;=\;\zeta$ 
for all elements $a$ of $G$.
Examples

The zero $0$ is the absorbing element for multiplication (or multiplicatively absorbing) in every ring $(R,\,+,\,\cdot)$.

The zero ideal $(0)$ is absorbing for ideal multiplication.

The zero vector $\vec{0}$ is the absorbing element for the vectoral multiplication “$\times$”.

The empty set $\varnothing$ is the absorbing element for the intersection operation “$\cap$” and also for the Cartesian product “$\times$”.

In an upper semilattice, an element is absorbing iff it is the top element. Dually, an element is absorbing iff it is the bottom element in a lower semilattice.
As the examples give reason to believe, the absorbing element for an operation is always unique. Indeed, if in addition to $\zeta$ we have in $G$ another absorbing element $\eta$, then we must have $\eta=\zeta\!*\!\eta=\zeta$.
Because $\zeta\!*\!\zeta=\zeta$, the absorbing element is idempotent.
Mathematics Subject Classification
20N02 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new correction: examples and OEIS sequences by fizzie
Oct 13
new correction: Define Galois correspondence by porton
Oct 7
new correction: Closure properties on languages: DCFL not closed under reversal by babou
new correction: DCFLs are not closed under reversal by petey
new question: Lorenz system by David Bankom
Oct 2
new correction: Many corrections by Smarandache
Sep 28
new question: how to contest an entry? by zorba
new question: simple question by parag
Sep 26
new question: Latent variable by adam_reith
Corrections
Also mentions by CWoo ✓
minor by CWoo ✓
absorbing elements in semilattices by CWoo ✓