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In [[mathematics]], an '''absorbing element''' is a special type of element of a [[Set (mathematics)|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 '''[[Semigroup#Identity and zero|zero element]]'''<ref>J.M. Howie, p. 2-3</ref><ref name=kkm>M. Kilp, U. Knauer, A.V. Mikhalev p. 14-15</ref> because there is no risk of confusion with [[zero element | other notions]] of zero. In this article the two notions are synonymous.  An absorbing element may also be called an '''annihilating element'''.  
 
== Definition ==
Formally, let (''S'', ∘) be a set ''S'' with a binary operation ∘ on it (known as a [[magma (algebra)|magma]]). A '''zero element''' is an element ''z'' such that for all ''s'' in ''S'', ''z''∘''s''=''s''∘''z''=''z''. A refinement<ref name=kkm/> are the notions of '''left zero''', where one requires only that ''z''∘''s''=''z'', and '''right zero''', where ''s''∘''z''=''z''.
 
Absorbing elements are particularly interesting for [[semigroup]]s, 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.<ref>J.S. Golan p. 67</ref>
 
==Properties==
* If a magma has both a left zero <math>z</math> and a right zero <math>z'</math>, then it has a zero, since <math>z = z \times z' = z'</math>.
* If a magma has a zero element, then the zero element is [[Uniqueness quantification|unique]].
 
==Examples==
*The most well known example of an absorbing element in algebra is multiplication, where any number multiplied by zero equals zero.  Zero is thus an absorbing element.
* [[Floating point]] arithmetics as defined in IEEE-754 standard contains a special value called Not-a-Number ("NaN"). It is absorbing element for every operation, ie. x+NaN=NaN+x=NaN, x-NaN=NaN-x=NaN etc.
* The set of [[binary relations]] over a set ''X'', together with the [[composition of relations]] forms a [[monoid]] with zero, where the zero element is the empty relation ([[empty set]]).
* The closed interval ''H''=<nowiki>[</nowiki>0, 1<nowiki>]</nowiki> with x∘y=min(x,y) is also a monoid with zero, and the zero element is 0.
* More examples:
{| class="wikitable" style="margin: 1em auto 1em auto"
!set!!operation!!absorber
|-
|[[real number]]s||· (multiplication)||[[0 (number)|0]]
|-
|[[nonnegative integer]]s||[[greatest common divisor]]||1
|-
|''n''-by-''n'' square [[matrix (mathematics)|matrices]]|| · (multiplication)
|[[zero matrix|matrix of all zeroes]]
|-
|[[extended real number]]s || minimum/infimum || −∞
|-
|[[extended real number]]s || maximum/supremum || +∞
|-
|[[Set (mathematics)|set]]s || ∩ (intersection) || { } ([[empty set]])
|-
|subsets of a set ''M'' || ∪ (union) || ''M''
|-
|[[boolean logic]] || ∧ ([[logical and]]) || ⊥ (falsity)
|-
|[[boolean logic]] || ∨ ([[logical or]]) || ⊤ (truth)
|}
 
==See also==
*[[Identity element]]
*[[Null semigroup]]
 
==Notes==
{{reflist|2}}
 
==References==
*{{cite book|last= Howie|first= John M.|title=Fundamentals of Semigroup Theory|year=1995|publisher=[[Clarendon Press]]|id=ISBN 0-19-851194-9}}
* 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, ISBN 3-11-015248-7.
* {{cite book |title=Semirings and Their Applications |first=Jonathan S. |last=Golan |year=1999 |publisher=Springer |isbn=0-7923-5786-8}}
 
==External links==
* [http://planetmath.org/encyclopedia/AbsorbingElement.html Absorbing element] at PlanetMath
 
[[Category:Semigroup theory]]
[[Category:Binary operations|*Absorbing element]]
 
__NOTOC__

Revision as of 17:41, 23 January 2014

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.

Definition

Formally, let (S, ∘) be a set S with a binary operation ∘ on it (known as a magma). A zero element is an element z such that for all s in S, z∘s=s∘z=z. A refinement[2] are the notions of left zero, where one requires only that z∘s=z, and right zero, where s∘z=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]

Properties

  • If a magma has both a left zero z and a right zero z′, then it has a zero, since z=z×z′=z′.
  • If a magma has a zero element, then the zero element is unique.

Examples

  • The most well known example of an absorbing element in algebra is multiplication, where any number multiplied by zero equals zero. Zero is thus an absorbing element.
  • Floating point arithmetics as defined in IEEE-754 standard contains a special value called Not-a-Number ("NaN"). It is absorbing element for every operation, ie. x+NaN=NaN+x=NaN, x-NaN=NaN-x=NaN etc.
  • The set of binary relations over a set X, together with the composition of relations forms a monoid with zero, where the zero element is the empty relation (empty set).
  • The closed interval H=[0, 1] with x∘y=min(x,y) is also a monoid with zero, and the zero element is 0.
  • More examples:
set operation absorber
real numbers · (multiplication) 0
nonnegative integers greatest common divisor 1
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

Notes

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References

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  • 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, ISBN 3-11-015248-7.
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links


  1. ↑ J.M. Howie, p. 2-3
  2. ↑ 2.0 2.1 M. Kilp, U. Knauer, A.V. Mikhalev p. 14-15
  3. ↑ J.S. Golan p. 67
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