Absorbing element

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.
Formally, let be a set "S" with a closed binary operation • on it (known as a magma). A zero element is an element "z" such that for all "s" in "S", . A refinement are the notions of left zero, where one requires only that , and right zero, where .
Absorbing elements are particularly interesting for semigroups, especially the multiplicative semigroup of a semiring. In the case of a semiring with 0, the definition of an absorbing element is sometimes relaxed so that it is not required to absorb 0; otherwise, 0 would be the only absorbing element.