An element $\zeta$ of a groupoid  $(G,\,*)$  is called an absorbing element (in French un élément absorbant) for the operation “$*$”, if it satisfies

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 “$\times$”.

• 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. 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.

If a group has an absorbing element, the group is trivial.