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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

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 Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...

). 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

s, especially the multiplicative semigroup of a

semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...

. 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 In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...

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 In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...

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 In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...

with zero, where the zero element is the

empty relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) 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 othe ...

). * 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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