PlanetMath: absorbing element

(more info)

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absorbing element

(Definition)

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

$\displaystyle \zeta\!*\!a = a\!*\!\zeta = \zeta$

for all elements

Examples

The zero 0 is the absorbing element for multiplication (or multiplicatively absorbing) in every ring $(R,\,+,\,\cdot)$ .
The zero ideal 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” is the absorbing element for the union operation “ $\cup$ ”:
$\displaystyle E\!\cup\!A = A\!\cup\!E = E \quad \forall A\subseteq E.$
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 another absorbing element $\eta$ , then we must have $\eta = \zeta\!*\!\eta = \zeta$ .