Absorbing element

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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 [1] [2] 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.

Contents

Definition

Formally, let (S, •) be a set S with a closed binary operation • on it (known as a magma). A zero element (or an absorbing/annihilating element) is an element z such that for all s in S, zs = sz = z. This notion can be refined to the notions of left zero, where one requires only that zs = z, and right zero, where sz = z. [2]

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. [3]

Properties

Examples

DomainOperationAbsorber
real numbers multiplication 0
integers greatest common divisor 1
n-by-n square matrices matrix multiplication matrix of all zeroes
extended real numbers minimum/infimum−∞
maximum/supremum+∞
sets intersection empty set
subsets of a set MunionM
Boolean logic logical and falsity
logical or truth

See also

Notes

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