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

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

• 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	2013-03-22 15:46:12
Last modified on	2013-03-22 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