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## Main Question or Discussion Point

Can someone please help me?

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Can someone please help me?

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Do you mean the axiom of an R-algebra? Or do you mean: the axioms of the discipline "algebra"?

Please provide some more background: what do you need to know it for? Where have you read it? What is your current knowledge? etc.

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hunt_mat

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Are you referring to an algrbra, a group, a ring or a field?

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Fredrik

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Really short answer: There are no axioms for algebra. There's just the axioms of set theory (and an associated proof theory), and then the axioms that define specific algebraic structures, such as groups, vector spaces, or the field of real numbers.

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hunt_mat

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The definition of an algebra I know is that of an algebra defined over a vector space.

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What are the axioms of algebra?

Wrong question...

What are the axioms of**an** algebra?

That's more like it!

I've only very recently, think yesterday, discovered that you really need to go to the

depths of logic to answer your question properly. Basically an algebra is a set with an

associated binary operation acting on it, lets <S,·> is the algebra where · is S x S → S.

Now, there is a far deeper thing going on here. In terms of logic S is simply a language of

symbols that have absolutely no meaning whatsoever but by certain axioms of logic we can

associate · with specific meaning that suits our desires. Most likely you were referring to

the axioms of group theory when you spoke of algebra and a group is just an algebraic

structure denoted <G,•,e> where • carries with it three specific group axioms that elements

of G must obey. We can describe different algebras or different structures if we want by

giving · different meaning or changing the arity of · etc...

So in short you want to know what algebra you are working in because what's really going

on is that you are working in an algebraic structure which is a specific case of a beast

called a "structure" and a structure itself is modelled on very specific rules in order to

make sense.

That's all I can say for now and may have gotten something wrong but will return to correct

it as I continue reading https://www.amazon.com/dp/0387942580/?tag=pfamazon01-20 treasure trove of a book :!!)

Wrong question...

What are the axioms of

That's more like it!

I've only very recently, think yesterday, discovered that you really need to go to the

depths of logic to answer your question properly. Basically an algebra is a set with an

associated binary operation acting on it, lets <S,·> is the algebra where · is S x S → S.

Now, there is a far deeper thing going on here. In terms of logic S is simply a language of

symbols that have absolutely no meaning whatsoever but by certain axioms of logic we can

associate · with specific meaning that suits our desires. Most likely you were referring to

the axioms of group theory when you spoke of algebra and a group is just an algebraic

structure denoted <G,•,e> where • carries with it three specific group axioms that elements

of G must obey. We can describe different algebras or different structures if we want by

giving · different meaning or changing the arity of · etc...

So in short you want to know what algebra you are working in because what's really going

on is that you are working in an algebraic structure which is a specific case of a beast

called a "structure" and a structure itself is modelled on very specific rules in order to

make sense.

That's all I can say for now and may have gotten something wrong but will return to correct

it as I continue reading https://www.amazon.com/dp/0387942580/?tag=pfamazon01-20 treasure trove of a book :!!)

Last edited by a moderator:

- #7

Fredrik

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So I think my answer above isn't very appropriate either. I think he's probably asking for the rules of

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Fredrik this is the linear and abstract algebra forum so it's a fair assumption on our partSo I think my answer above isn't very appropriate either. I think he's probably asking for the rules ofelementaryalgebra, i.e. the kind of stuff you're allowed to do with variables that represent real numbers. But it looks like the OP has abandoned the thread, so we will probably never know.

that the OP is not just looking for the field axioms or just the laws of arithmetic but you're

probably right I doubt we'll find out :tongue2:

I can only quote it from memory but on Saturday I was looking in Suppes Introduction to@sponsoredwalk: Sounds like you're going for the definition of "algebra" from universal algebra, and not getting it right. (Why are you only including a binary operation?)

Logic in the library & he made it absolutely clear that an algebra is a non-empty set A and

a binary operation •. Furthermore he qualifies himself by calling this binary operation a

function and in this link the author also qualifies himself by pointing out that an algebraic

structure is a structure (the logical concept) that has functions but no relations.

edit: page 183 also mentions this

Also:

So a binary operation is simply one of the many functions that can be used. PerhapsSince the signatures that arise in algebra often contain only function symbols, a signature

with no relation symbols is called an algebraic signature. A structure with such a signature

is also called an algebra; this should not be confused with the notion of an algebra over a

field.

http://en.wikipedia.org/wiki/Structure_(mathematical_logic)

Suppes explains this later on idk

So I don't think I was wrong just maybe a little restricted in quoting Suppes instead

of the more general idea.

The Ebbinghaus book I quoted in the link in my post also makes it clear that this is all

part of logic and if you read the beginning of https://www.amazon.com/dp/0521168481/?tag=pfamazon01-20 in the preview or

on googlebooks you'll see that things are constructed this way to satiate a formalist & leave

no room for error.

Universal Algebra is a generalization of these logical concepts but in no way is the universal

algebra definition - which is simply a generalization of these concepts - somehow distinct

from the rest of mathematics and it really can't be if these concepts are the very beginning

of the semantics of the syntax of first-order logic. As far as I understand it this is just

what's really going on underneath all of the shortcuts & notational conveniences we employ

but I am not certain & judging by my experiences over the past few months I'll probably be

fed new facts soon contradicting most of this

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