In
mathematics, an absorbing element (or annihilating element) is a special type of element of a
set with respect to a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...
theory, the absorbing element is called a
zero element
In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive iden ...
[M. Kilp, U. Knauer, A.V. Mikhalev pp. 14–15] 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
Formally, let be a set ''S'' with a closed binary operation • on it (known as a
magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...
). A zero element is an element ''z'' such that for all ''s'' in ''S'', . This notion can be refined to the notions of left zero, where one requires only that , and right zero, where .
[
Absorbing elements are particularly interesting for ]semigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...
s, especially the multiplicative semigroup of a semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
The term rig is also used occasionally—this originated as a joke, suggesting that rigs