Absorbing element

Absorbing element

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 [J.M. Howie, p. 2-3] M. Kilp, U. Knauer, A.V. Mikhalev p. 14-15] because there is no risk of confusion with other notions of zero. In this article the two notions are synonymous.

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 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. [J.S. Golan p. 67]

Properties

* If a magma has both a left and a right zero, then it has a zero.
* If a magma has a zero element, then the zero element is unique.

Examples

* 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.

ee also

*Identity element

Notes

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 3110152487.
*

External links

* [http://planetmath.org/encyclopedia/AbsorbingElement.html Absorbing element] at PlanetMath

__NOTOC__


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Absorbing set — In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space. Alternative terms are radial or absorbent set.DefinitionGiven a vector space X …   Wikipedia

  • Zero element — In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context:*An additive identity, i.e., the… …   Wikipedia

  • Inverse element — In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can undo the effect of combination with… …   Wikipedia

  • Identity element — In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related… …   Wikipedia

  • chemical element — Introduction also called  element,         any substance that cannot be decomposed into simpler substances by ordinary chemical processes. Elements are the fundamental materials of which all matter is composed.       This article considers the… …   Universalium

  • rare-earth element — /rair errth /, Chem. any of a group of closely related metallic elements, comprising the lanthanides, scandium, and yttrium, that are chemically similar by virtue of having the same number of valence electrons. Also called rare earth metal. [1955 …   Universalium

  • Semigroup — This article is about the algebraic structure. For applications to differential equations, see C0 semigroup. In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup… …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • 0 (number) — Zero redirects here. For other uses, see Zero (disambiguation). 0 −1 0 1 2 3 4 5 6 7 8 …   Wikipedia

  • Converse nonimplication — In logic, converse nonimplication is a logical connective which is the negation of the converse of implication. Contents 1 Definition 1.1 Truth table 1.2 Venn diagram …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”