In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation zero may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous.

Definition

on it (known as a magma). A zero element (or an absorbing/annihilating element. This notion can be refined to the notions of left zero, and right zero.

Absorbing elements are particularly interesting for semigroups, especially the multiplicative semigroup of a semiring would be the only absorbing element.

Properties

• .

• A magma can have at most one zero element.

Examples

• The most well known example of an absorbing element comes from elementary algebra, where any number multiplied by zero equals zero. Zero is thus an absorbing element.

• The zero of any ring. This property holds true also in a rng since multiplicative identity isn't required.

• Floating point, etc.

• The set of binary relations, together with the composition of relations forms a monoid with zero, where the zero element is the empty relation (empty set).

• .

• More examples:

See also

• Ideal (ring theory)

Notes

References

External links

• Absorbing element at PlanetMath