en>Yobot: Tagging using AWB (10703)

2015-01-07T15:15:03Z

Tagging using AWB (10703)

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	~~{{ other uses2\|Zero}}~~		The author's title is Christy. I am presently a journey agent. My wife and I reside in Mississippi but now I'm considering other options. To perform lacross is one of the things she loves most.<br><br>Stop by my weblog ... [http://modenpeople.co.kr/modn/qna/292291 phone psychic]

	In [[mathematics]], a '''zero element''' is one of several generalizations of [[0 (number)\|the number zero]] to other [[algebraic structure]]s. These alternate meanings may or may not reduce to the same thing, depending on the context.

	~~==Additive identities==~~
	~~An '''[[additive identity]]''' is the [[identity element]] in an [[Abelian group\|additive group]]. It generalises the property {{nowrap\|1=0 + ''x'' = ''x''}}. Examples include:~~
	*The '~~''[[zero vector]]''' under [[vector addition]]~~
	*The '''zero function''' or '''zero map''' under function addition {{nowrap\|1=(''f'' + ''g'')(''x'') = ''f''(''x'') + ''g''(''x'')}}, defined by {{nowrap\|1=''z''(''x'') = 0}}, since {{nowrap\|1=''z'' + ''f'' = ''f''}}
	*The '''[[empty set]]''' under [[Union (set theory)\|set union]]
	*An '''[[empty sum]]''' or '''empty [[coproduct]]'''
	*An '''[[Initial and terminal objects\|initial object]]''' in a [[category (mathematics)\|category]] (an empty coproduct, and so an identity under [[coproduct]]s)

	~~==Absorbing elements==~~
	~~An '''[[absorbing element]]''' in a multiplicative [[semigroup]] or [[semiring]] generalises the property {{nowrap\|1=0 × ''x'' = 0}}. Examples include:~~
	*The '''[[empty set]]''', which is ~~an absorbing element under [[Cartesian product]] of sets, since {{nowrap\|1={ } × ''S'' = { }<nowiki/>}}~~
	*The '''zero function''' or '''zero map''' under function multiplication {{nowrap\|1=(''f'' × ''g'')(''x'') = ''f''(''x'') × ''g''(''x'')}}, defined by {{nowrap\|1=''z''(''x'') = 0}}, since {{nowrap\|1=''z'' × ''f'' = ''z''}}
	~~Many absorbing elements are also additive identities, including the empty set and the zero function~~. ~~Another important example is the distinguished element 0 in~~ a '''[[field (mathematics)\|field]]''' or '''[[ring (mathematics)\|ring]]''', which is both the additive identity and the multiplicative absorbing element, and whose [[principal ideal]] is the smallest ideal.

	~~==Zero objects==~~
	~~A '''zero object''' in a [[category (mathematics)\|category]] is both an [[Initial and terminal objects\|initial and terminal object]] (and so an identity under both [[coproduct]]s~~ and ~~[[Product (category theory)\|products]]). For example, the trivial structure (containing only the identity) is a zero object~~ in ~~categories where morphisms must map identities to identities. Specific examples include:~~
	*The '~~''[[trivial group]]''', containing only the identity (a zero object in the [[category of groups]])~~
	*The '''zero module''', containing only the identity (a zero object in the category of [[module (mathematics)\|modules]] over a ring)

	~~==Zero morphisms==~~
	~~A '''[[zero morphism]]''' in a [[category (mathematics)\|category]] is a generalised absorbing element under [[function composition]]: any morphism composed with a zero morphism gives a zero morphism~~. ~~Specifically, if {{nowrap\|0<sub>''XY''</sub> : ''X'' → ''Y''}}~~ is the zero morphism among morphisms from ''X'' to ''Y'', and {{nowrap\|''f'' : ''A'' → ''X''}} and {{nowrap\|''g'' : ''Y'' → ''B''}} are arbitrary morphisms, then {{nowrap\|''g'' ∘ 0<sub>''XY''</sub> {{=}} 0<sub>''XB''</sub>}} and {{nowrap\|0<sub>''XY''</sub> ∘ ''f'' {{=}} 0<sub>''AY''</sub>}}.

	If a category has a zero object '''0''', then there are canonical morphisms {{nowrap\|''X'' → '''0'''}} and {{nowrap\|'''0''' → ''Y'',}} and composing them gives a zero morphism {{nowrap\|0<sub>''XY''</sub> : ''X'' → ''Y''}}. In the [[category of ~~groups]], for example, zero morphisms are morphisms which always return group identities, thus generalising~~ the ~~function {{nowrap\|''z''(''x'') {{=}} 0.}}~~

	~~==Least elements==~~
	~~A '''[[least element]]''' in a [[partially ordered set]] or [[Lattice (order)\|lattice]] may sometimes be called a zero element, and written either as 0 or ⊥~~.

	~~==Zero module==~~
	In [[mathematics]], the '''zero module''' is the [[module (mathematics)\|module]] consisting of only the additive [[identity element\|identity]] for the module's [[addition]] function. In the [[integer]]s, this identity is [[0 (number)\|zero]], which gives the name ''zero module''. That the zero module is in fact a module is simple to show; it is closed under addition and [[multiplication]] trivially.

	~~==Zero ideal==~~
	~~In [[mathematics]], the '''zero ideal''' in a [[Ring (mathematics)\|ring]]~~ <~~math~~>R<~~/math~~> ~~is the ideal <math>\{ 0 \}</math> consisting of only the additive identity (or [[0 (number)\|zero]] element)~~. ~~It is immediate to show that this is an [[ideal (ring theory)\|ideal]]~~.

	~~==Zero matrix==~~
	~~In [[mathematics]], particularly [[linear algebra]], a '''zero matrix''' is a [[matrix (mathematics)\|matrix]] with all its entries being [[0 (number)\|zero]]~~. ~~Some examples of zero matrices are~~

	~~:<math>~~
	~~0_{1,1} = \begin{bmatrix}~~
	~~0 \end{bmatrix}~~
	,\
	~~0_{2,2} = \begin{bmatrix}~~
	~~0 & 0 \\~~
	~~0 & 0 \end{bmatrix}~~
	,\
	~~0_{2,3} = \begin{bmatrix}~~
	~~0 & 0 & 0 \\~~
	~~0 & 0 & 0 \end{bmatrix}~~
	,\
	~~</math>~~

	~~The set of ''m''×''n'' matrices with entries in a~~ [[ring (mathematics)\|ring]] K forms a module <math>K_{m,n} \,</math>. The zero matrix <math>0_{K_{m,n}} \, </math> in <math>K_{m,n} \, </math> is the matrix with all entries equal to <math>0_K \, </math>, where <math>0_K \, </math> is the additive identity in K.

	:~~<math>~~
	~~0_{K_{m,n}} = \begin{bmatrix}~~
	~~0_K & 0_K & \cdots & 0_K \\~~
	~~0_K & 0_K & \cdots & 0_K \\~~
	~~\vdots & \vdots & & \vdots \\~~
	~~0_K & 0_K & \cdots & 0_K \end{bmatrix}~~
	</~~math>~~

	~~The zero matrix is the additive identity in <math>K_{m,n} \, <~~/~~math>~~. ~~That is, for all <math>A \in K_{m,n} \, </math> it satisfies~~

	~~:<math>0_{K_{m,n}}+A = A + 0_{K_{m,n}} = A</math>~~

	There is exactly one zero matrix of any given size ''m''×''n'' having entries in a given ring, so when the context is clear one often refers to ''the'' zero matrix. In general the zero element of a ring is unique and typically denoted as 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.

	~~The zero matrix represents the [[linear transformation]] sending all vectors to the [[zero vector]]~~.

	~~==Zero tensor==~~
	~~In [[mathematics]], the '''zero tensor''' is a [[tensor]], of any order, all of whose components are [[0 (number)\|zero]]. The zero tensor of order 1 is sometimes known as the [[zero vector]].~~

	~~Taking a [[tensor product]] of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.~~

	~~==Zero divisor==~~
	~~A '''[[zero divisor]]''' in a [[ring (mathematics)\|ring]] ''R'' is a non-zero element ''a'' ∈ ''R'' such that ''ab'' = 0 for some non-zero ''b'' ∈ ''R''.~~

	~~==Zerosumfree monoid==~~
	~~In [[abstract algebra]], an additive [[monoid]] <math>(M, 0, +)<~~/~~math> is said to be '''zerosumfree''' if nonzero elements do not sum to zero. Formally:~~

	~~:<math>(\forall a,b\in M)\ a + b = 0 \implies a = 0 = b \!<~~/~~math>~~

	~~This means that the only way zero can be expressed as a sum is as <math>0 + 0<~~/~~math>.~~

	~~==See also==~~

	*[[Zero object]]
	*[[Zero of a function]]
	*[[Zero]] non mathematical uses.

	~~[[Category:Zero]~~]

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q369367

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Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q369367

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en>BattyBot: merging templates into Template:Multiple issues & general fixes using AWB (7961)

2012-02-21T03:59:48Z

merging templates into Template:Multiple issues & general fixes using AWB (7961)

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en>Yobot: Tagging using AWB (10703)

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q369367

en>BattyBot: merging templates into Template:Multiple issues & general fixes using AWB (7961)