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

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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 (http://planetmath.org/IdealMultiplicationLaws).

•
The zero vector $\overrightarrow{0}$ is the absorbing element for the vectoral multiplication (http://planetmath.org/CrossProduct) “$\times $”.

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

•
The “universal set” $E$ is the absorbing element for the union operation “$\cup $”:
$$E\cup A=A\cup E=E\mathit{\hspace{1em}}\forall A\subseteq E.$$ 
•
In an upper semilattice^{}, an element is absorbing iff it is the top element (http://planetmath.org/BoundedLattice). Dually, an element is absorbing iff it is the bottom element (http://planetmath.org/BoundedLattice) in a lower semilattice^{}.
As the examples give reason to believe, the absorbing element for an operation is always unique. Indeed, if in 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^{}.
If a group has an absorbing element, the group is trivial (http://planetmath.org/Subgroup^{}).
Title  absorbing element 
Canonical name  AbsorbingElement 
Date of creation  20130322 15:46:12 
Last modified on  20130322 15:46:12 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  20 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 20N02 
Synonym  absorbant 
Synonym  absorbing 
Related topic  RingOfSets 
Related topic  ZeroElements 
Related topic  0cdotA0 
Related topic  AbsorbingSet 
Related topic  IdentityElementIsUnique 