What are the axioms of algebra?

[asap9993]

Main Question or Discussion Point

Can someone please help me?

 

[micromass]

Can you please elaborate on your question. Because it is hard to see what you mean...

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.

 

[hunt_mat]

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

 

[Fredrik]

Sounds like he's referring to the axioms for the branch of mathematics called "algebra". I would also like to know a bit more about the OP's mathematical background before I attempt a detailed answer.

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.

 

[hunt_mat]

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

 

[sponsoredwalk]

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 :!!)

 

[Fredrik]

@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?) I think that's very far from what this person wanted to know. I think a person who asks for the "axioms of algebra" has never even heard of abstract algebra, or even groups. And universal algebra is of course something even more abstract (and known to a lot less people) than abstract algebra.

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

 

[sponsoredwalk]

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

Fredrik this is the linear and abstract algebra forum so it's a fair assumption on our part
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:

@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?)

I can only quote it from memory but on Saturday I was looking in Suppes Introduction to
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:

Since 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)

So a binary operation is simply one of the many functions that can be used. Perhaps
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