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In mathematics, the conventional definition of the concept of [[vector space]] relies upon the [[algebra]]ic concept of a [[field (mathematics)|field]]. This article treats an algebraic definition that does not require that concept. If vector spaces are redefined as [[universal algebra|(universal) algebras]] as below, no preliminary introduction of fields is necessary. On the contrary, fields will come from such '''vector space algebras'''. | |||
One of the ways to do this is to keep the first four [[Abelian group]] axioms on addition in the standard formal definition and to formalize its [[homothety|''geometric'' idea of scaling]] only by notions that concern every universal algebra.<ref>The “generalized conception of space” in the preface of {{harvtxt|Whitehead|1898}} is the first published claim that geometric ideas concern Universal Algebra too. It expands a similar claim in the preface of {{harvtxt|Fibonacci|1202}}.</ref> Vector space algebras consist<!-- this vagueness is related to the fact that an algebra can be defined either by a set of operations or by an indexing of them depending on its applications --> of ''one'' [[binary operation]] "<!-- avoiding a sea of blue -->[[addition|+]]" and of ''unary'' operations {{mvar|δ}}, which form a ''nonempty'' set {{math|Δ}}, that satisfy the following conditions, which do not involve fields. | |||
# (''Total homogeneous algebra'') There is a single set {{mvar|V}} such that every operation takes its two arguments or its argument from the whole {{mvar|V}} and gives its value in it. | |||
# (''Abelian group'') + satisfies the above mentioned axioms. | |||
# (''Basis dilation'') There is a [[basis (universal algebra)|basis set]] <math>B\subseteq V</math> such that, for every such {{mvar|δ}} that is not constant, all the values <math>\delta(b)</math>, where {{mvar|b}} ranges over {{mvar|B}}, again form a basis set. | |||
# (''Dilatations'') {{math|Δ}} is the set of functions {{mvar|δ}} that satisfy the previous conditions and ''preserve'' all operations, namely <math>\delta(v+w)=\delta(v)+\delta(w)</math> and <math>\delta(\gamma(v))=\gamma(\delta(v))</math>, for all <math>\gamma\in\Delta</math> and all <math>v,w\in V</math>. | |||
{{harvtxt|Ricci|2008}} proves that these vector space algebras are the very universal algebras that any standard vector space defines by its addition and the [[scalar multiplication]]s by any given scalar (namely each <math>a\in F</math> gets a <math>\delta\in\Delta</math> such that <math>\delta(v)=av</math>). {{harvtxt|Ricci|2007}} proves that every such a universal algebra defines a suitable field. (Hence, it proves that these conditions imply all the axioms of the standard [[vector space|formal definition]], as well as all the defining properties in [[field (mathematics)|definition 3 of a field]].)<ref>This construction also concern the [[zero vector space|space with only one (null) vector]], where the defined field [[trivial ring|only has the zero as element]]. This slightly extends definition 3 of a field, which only considers fields with at least two elements.</ref> | |||
Since the field is defined from such vector space algebra, this is an ''algebraic'' construction of fields, which is an instance of a more general algebraic construction: the "[[endowed dilatation monoid]]" {{harv|Ricci|2007}}. However, as far as fields are concerned, there also is a ''geometric'' construction. Chapter II in {{harv|Artin|1957}} shows how to get them starting from geometric axioms concerning points and lines. | |||
== Notes == | |||
{{reflist}} | |||
== References == | |||
# {{cite book |ref=harv |last=Artin |first=E. |authorlink=Emil Artin |date=1957 |title=Geometric Algebra |publisher=Interscience Publishers |location=New York}} | |||
# {{cite book |ref=CITEREFFibonacci1202 |last=Fibonacci |first=L. |authorlink=Fibonacci |origyear=1202 |date=1857 |title=Liber Abbaci |publisher=reprinted by Tipografia delle Scienze Matematiche e Fisiche, B. Boncompagni ed. |location=Rome}} | |||
# {{cite journal |ref=harv |last=Ricci |first=G. |date=2007 |title=Dilatations kill fields |journal=International Journal of Mathematics, Game Theory and Algebra |volume=16 |issue=5/6 |pages=13–34}} | |||
# {{cite book |ref=harv |last=Ricci |first=G. |date=2008 |chapter=Another characterization of vector spaces without fields |editor-last=G. Dorfer, G. Eigenthaler, H. Kautschitsch, W. More, W.B. Müller (Hrsg.) |title=Contributions to General Algebra |volume=18 |publisher=Verlag Johannes Heyn GmbH & Co |location=Klagenfurt |date=April 2008 |pages=139–150 |isbn=978-3-7084-0303-8}} | |||
# {{cite book |ref=harv |last=Whitehead |first=A.N. |authorlink=Alfred North Whitehead |date=1898 |title=A treatise on Universal Algebra with applications |volume=1 |publisher=Cambridge University Press |location=Cambridge}} | |||
[[Category:Vectors]] | |||
[[Category:Universal algebra]] | |||
[[Category:Linear algebra]] | |||
[[Category:Abstract algebra]] | |||
Latest revision as of 13:53, 22 December 2013
In mathematics, the conventional definition of the concept of vector space relies upon the algebraic concept of a field. This article treats an algebraic definition that does not require that concept. If vector spaces are redefined as (universal) algebras as below, no preliminary introduction of fields is necessary. On the contrary, fields will come from such vector space algebras.
One of the ways to do this is to keep the first four Abelian group axioms on addition in the standard formal definition and to formalize its geometric idea of scaling only by notions that concern every universal algebra.[1] Vector space algebras consist of one binary operation "+" and of unary operations Template:Mvar, which form a nonempty set Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
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If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., that satisfy the following conditions, which do not involve fields.
- (Total homogeneous algebra) There is a single set Template:Mvar such that every operation takes its two arguments or its argument from the whole Template:Mvar and gives its value in it.
- (Abelian group) + satisfies the above mentioned axioms.
- (Basis dilation) There is a basis set such that, for every such Template:Mvar that is not constant, all the values , where Template:Mvar ranges over Template:Mvar, again form a basis set.
- (Dilatations) Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the set of functions Template:Mvar that satisfy the previous conditions and preserve all operations, namely and , for all and all .
Template:Harvtxt proves that these vector space algebras are the very universal algebras that any standard vector space defines by its addition and the scalar multiplications by any given scalar (namely each gets a such that ). Template:Harvtxt proves that every such a universal algebra defines a suitable field. (Hence, it proves that these conditions imply all the axioms of the standard formal definition, as well as all the defining properties in definition 3 of a field.)[2]
Since the field is defined from such vector space algebra, this is an algebraic construction of fields, which is an instance of a more general algebraic construction: the "endowed dilatation monoid" Template:Harv. However, as far as fields are concerned, there also is a geometric construction. Chapter II in Template:Harv shows how to get them starting from geometric axioms concerning points and lines.
Notes
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References
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- ↑ The “generalized conception of space” in the preface of Template:Harvtxt is the first published claim that geometric ideas concern Universal Algebra too. It expands a similar claim in the preface of Template:Harvtxt.
- ↑ This construction also concern the space with only one (null) vector, where the defined field only has the zero as element. This slightly extends definition 3 of a field, which only considers fields with at least two elements.