mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 theory, the absorbing element is called a zero elementM. 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). 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 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.J.S. Golan p. 67

Properties

* If a magma has both a left zero ''z'' and a right zero ''z''′, then it has a zero, since . * A magma can have at most one zero element.

Examples

*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 any ring is also an absorbing element. For an element ''r'' of a ring ''R'', ''r0=r(0+0)=r0+r0'', so ''0=r0'', as zero is the unique element ''a'' for which ''r-r=a'' for any ''r'' in the ring ''R''. This property holds true also in a rng 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., , , etc. * The set of binary relations over a set ''X'', together with the

composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...

forms a monoid with zero, where the zero element is the empty relation (

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

). * The closed interval with is also a monoid with zero, and the zero element is 0. * More examples:

Notes

References

* * M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, . * {{cite book , title=Semirings and Their Applications , first=Jonathan S. , last=Golan , year=1999 , publisher=Springer , isbn=0-7923-5786-8

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