absorbing element

An element ζ of a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath(G,*)  is called an absorbing element (in French un élément absorbant) for the operationMathworldPlanetmath*”, if it satisfies


for all elements a of G.


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

  • The zero idealMathworldPlanetmathPlanetmath (0) is absorbing for ideal multiplication (http://planetmath.org/IdealMultiplicationLaws).

  • The zero vector 0 is the absorbing element for the vectoral multiplication (http://planetmath.org/CrossProduct) “×”.

  • The empty setMathworldPlanetmath is the absorbing element for the intersectionMathworldPlanetmathPlanetmath operation “” and also for the Cartesian productMathworldPlanetmath×”.

  • The “universal setE is the absorbing element for the union operation “”:

  • In an upper semilatticePlanetmathPlanetmath, 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 semilatticePlanetmathPlanetmath.

As the examples give reason to believe, the absorbing element for an operation is always unique.  Indeed, if in to ζ we have in G another absorbing element η, then we must have  η=ζ*η=ζ.

Because  ζ*ζ=ζ,  the absorbing element is idempotentPlanetmathPlanetmath.

If a group has an absorbing element, the group is trivial (http://planetmath.org/SubgroupMathworldPlanetmathPlanetmath).

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