Inmathematics, anabsorbing element (orannihilating element) is a special type of element of aset with respect to abinary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. Insemigroup theory, the absorbing element is called azero element^[1][2] because there is no risk of confusion withother notions of zero, with the notable exception: under additive notationzero may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous.

Formally, let(S, •) be a setS with a closed binary operation • on it (known as amagma). Azero element (or anabsorbing/annihilating element) is an elementz such that for alls inS,z •s =s •z =z. This notion can be refined to the notions ofleft zero, where one requires only thatz •s =z, andright zero, wheres •z =z.^[2]

Absorbing elements are particularly interesting forsemigroups, especially the multiplicative semigroup of asemiring. 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]

• If a magma has both a left zeroz and a right zeroz′, then it has a zero, sincez =z •z′ =z′.
• A magma can have at most one zero element.

• 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 anyring is also an absorbing element. For an elementr of a ringR,r0 =r(0 + 0) =r0 +r0, so0 =r0, as zero is the unique elementa for whichr −r =a for anyr in the ringR. This property holds true also in arng since multiplicative identity isn't required.
• Floating point arithmetics as defined in IEEE-754 standard contains a special value called Not-a-Number ("NaN"). It is an absorbing element for every operation; i.e.,x + NaN = NaN +x = NaN,x − NaN = NaN −x = NaN, etc.
• The set ofbinary relations over a setX, together with thecomposition of relations forms amonoid with zero, where the zero element is theempty relation (empty set).
• The closed intervalH = [0, 1] withx •y = min(x,y) is also a monoid with zero, and the zero element is 0.
• More examples:

• Absorbing set – Set that can be "inflated" to reach any point
• Annihilator (disambiguation)
• Annihilator (ring theory) – Ideal that maps to zero a subset of a module
• Idempotent (ring theory) – In mathematics, element that equals its square – an elementx of a ring such thatx² =x
• Identity element – Specific element of an algebraic structure
• Null semigroup

• Howie, John M. (1995).Fundamentals of Semigroup Theory.Clarendon Press.ISBN 0-19-851194-9.
• Kilp, M.; Knauer, U.; Mikhalev, A.V. (2000), "Monoids, Acts and Categories with Applications to Wreath Products and Graphs",De Gruyter Expositions in Mathematics,29, Walter de Gruyter,ISBN 3-11-015248-7
• Golan, Jonathan S. (1999).Semirings and Their Applications. Springer.ISBN 0-7923-5786-8.

• Absorbing element at PlanetMath

• Semigroup theory
• Binary operations
• Algebraic properties of elements

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