View source for Absorbing element

{{Short description|Special type of element of a set with respect to a binary operation on that set.}}
In [[Mathematics|mathematics]], an '''absorbing element''' (or '''annihilating element''') is a special type of element of a [[Set (mathematics)|set]] with respect to a [[Binary operation|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|semigroup]] theory, the absorbing element is called a '''[[Semigroup#Identity and zero|zero element]]'''{{sfn|ps=|Howie|1995|pp=2–3}}{{sfn|ps=|Kilp|Knauer|Mikhalev|2000|pp=14–15}} because there is no risk of confusion with [[Zero element|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 {{nowrap|(''S'', •)}} be a set ''S'' with a closed binary operation • on it (known as a [[Magma (algebra)|magma]]). A '''zero element''' is an element ''z'' such that for all ''s'' in ''S'', {{nowrap|1=''z'' • ''s'' = ''s'' • ''z'' = ''z''}}. This notion can be refined to the notions of '''left zero''', where one requires only that {{nowrap|1=''z'' • ''s'' = ''z''}}, and '''right zero''', where {{nowrap|1=''s'' • ''z'' = ''z''}}.{{sfn|ps=|Kilp|Knauer|Mikhalev|2000|pp=14–15}}

Absorbing elements are particularly interesting for [[Semigroup|semigroup]]s, especially the multiplicative semigroup of a [[Semiring|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.{{sfn|ps=|Golan|1999|p=67}}

== Properties ==
* If a magma has both a left zero ''z'' and a right zero ''z''′, then it has a zero, since {{nowrap|1=''z'' = ''z'' • ''z''′ = ''z''′}}.
* 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 (mathematics)|ring]] is also an absorbing element. For an element ''r'' of a ring ''R'', {{nowrap|1=''r''0 = ''r''(0 + 0) = ''r''0 + ''r''0}}, so {{nowrap|1=0 = ''r''0}}, as zero is the unique element ''a'' for which {{nowrap|1=''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., {{nowrap|1=''x'' + NaN = NaN + ''x'' = NaN}}, {{nowrap|1=''x'' − NaN = NaN − ''x'' = NaN}}, etc.
* The set of binary relations over a set ''X'', together with the [[Composition of relations|composition of relations]] forms a [[Monoid|monoid]] with zero, where the zero element is the empty relation ([[Empty set|empty set]]).
* The closed interval {{nowrap|1=''H'' = [0, 1]}} with {{nowrap|1=''x'' • ''y'' = min(''x'', ''y'')}} is also a monoid with zero, and the zero element is&nbsp;0.
* More examples:
{| class="wikitable" style="margin: 1em auto 1em auto"
! Domain
! colspan=2 | Operation
! colspan=2 | Absorber
|-
| [[Real number|real number]]s
| ⋅ || multiplication
| 0 ||
|-
| [[Integer|integer]]s
|   || greatest common divisor
| 1 ||
|-
| ''n''-by-''n'' square [[Matrix (mathematics)|matrices]]
| || [[Matrix multiplication|matrix multiplication]]
| || [[Zero matrix|matrix of all zeroes]]
|-
| rowspan=2 | [[Extended real number line|extended real number]]s
|    || minimum/infimum
| −∞ ||
|-
|    || maximum/supremum
| +∞ ||
|-
| [[Set (mathematics)|set]]s
| ∩ || intersection
| ∅ || [[Empty set|empty set]]
|-
| subsets of a set ''M''
| ∪     || union
| ''M'' ||
|-
| rowspan=2 | [[Boolean algebra|Boolean logic]]
| ∧ || [[Logical conjunction|logical and]]
| ⊥ || falsity
|-
| ∨ || [[Logical disjunction|logical or]]
| ⊤ || truth
|}

== See also ==
* [[Idempotent (ring theory)]]{{snd}}an element ''x'' of a ring such that {{nowrap|1=''x''<sup>2</sup> = ''x''}}
* [[Identity element]]
* [[Null semigroup]]

== Notes ==
{{reflist}}

== References ==
* {{cite book
  |last= Howie |first= John M.
  |year=1995
  |title=Fundamentals of Semigroup Theory
  |publisher=Clarendon Press
  |isbn=0-19-851194-9
  }}
* {{citation
  |last1=Kilp |first1=M. 
  |last2=Knauer |first2=U.
  |last3=Mikhalev |first3=A.V.
  |year=2000
  |title=Monoids, Acts and Categories with Applications to Wreath Products and Graphs
  |journal=De Gruyter Expositions in Mathematics
  |volume=29
  |publisher=Walter de Gruyter
  |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/absorbingelement Absorbing element] at PlanetMath

[[Category:Semigroup theory]]
[[Category:Binary operations|*Absorbing element]]

{{Sourceattribution|Absorbing element}}