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

• The empty set $\varnothing$ 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 “$\cup$”:

 $E\!\cup\!A\;=\;A\!\cup\!E\;=\;E\quad\forall A\;\subseteq\;E.$
• 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