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.
• 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.