Absorbing element

In mathematics, an absorbing 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. In this article the two notions are synonymous. An absorbing element may also be called an annihilating element.


Formally, let (S, •) 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, zs = sz = z. A refinement[2] are the notions of left zero, where one requires only that zs = z, and right zero, where sz = z.

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]



Set Operation Absorber
Real numbers ⋅ (multiplication)0
Integersgreatest common divisor1
n-by-n square matrices ⋅ (multiplication) Matrix of all zeroes
Extended real numbers minimum/infimum −∞
Extended real numbers maximum/supremum +∞
Sets ∩ (intersection) { } (empty set)
Subsets of a set M ∪ (union) M
Boolean logic ∧ (logical and) ⊥ (falsity)
Boolean logic ∨ (logical or) ⊤ (truth)

See also


  1. J.M. Howie, pp. 2–3
  2. 1 2 M. Kilp, U. Knauer, A.V. Mikhalev pp. 14–15
  3. J.S. Golan p. 67


External links

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