Can you imagine a group with infinitely many identity elements? A structure like this have discovered mathematician Algirdas Javtokas, he calls it a beta group. I have read his axioms in Beta Algebra book, but still it is hard to understand how such structure can “work”. Maybe you know something more about this?

Can you imagine a group with infinitely many identity elements?

No I can´t but I have the slight feeling that this is because an identity element is not what I imagine it to be. For me, 1 is an identity element exaclty if 1A = A1 = A for any A. For this, there should imho not exist more than one for each group: Assume A and B are two identity elements: A = AB = B.

Can you imagine a group with infinitely many identity elements?

 

no, since a group, by definition has a unique identitiy element. You might want to notice that whatever you refer to is strictly not called a group even by the person who discovered/invented it.

no, since a group, by definition has a unique identitiy element.

 

OK, now I'm confused. I was taught that the definition of a group states the existence of the identity element, and that a theorem establishes the uniqueness of the identity element. Is that not the mainstream view? If not, where am I going wrong?

For this, there should imho not exist more than one for each group: Assume A and B are two identity elements: A = AB = B.

a group' date=' by definition has a unique identitiy element. [/quote']

the definition of a group states the existence of the identity element, and that a theorem establishes the uniqueness of the identity element.

I actually wonder what your confusion is about Tom. To me it seems that we were all saying the same thing. I assume your confusion arises from the term "by definition" used by Matt? Not being a native english speaker I´d interpret "by definition" as both "is included in the definition" and "is a direct result of the definition". As I tried to show, the uniqueness of the 1 is garanteed by the set of conditions for a group I know. Adding an additional (superfluous) condition that there is a unique 1 will not change anything as both sets of conditions will still lead to the same definition in the sense that the set of all possible objects being a group doesn´t change.

Or in other words: I don´t think that a mathematical definition for something must behave like a base for a vector space (minimal set of generating vectors) but more like a generating system (vector space is clearly defined).

It perhaps wasn't the best use of the word 'definition'. Better would be, perhaps: it is a simple consequence of the axioms of a group that the identity is unique. Or indeed any other algebraic structure that has elements i and j that satisfy ix=x=xi = jx=xj for all x must have that i=j.

If you actually read the paper that google leads you to you see that all the author has done is (naturally) redefine the word 'identity' so that it only has to act as 1 on some subset of the underlying set of the 'beta group'. Normally we would call that an idempotent.

 

It is rather trivial to construct formal examples of them, whether or not they are of any use is another question entirely, or if they are indeed internally consistent. Noticeably there are no concrete examples of them given, nor am I entirely happy with his demonstration of internal consistency.

 

http://uosis.mif.vu.lt/%7ejavtokas/files/betagroups.pdf

 

BG3 contains the worrying phrase

 

for e_i there are e_j such that e_ie_j=e_j for all i,j in N.

 

which is confusingly written. I suspect he is trying to say that there are a countable number of e_j satisfying e_i=e_j.

 

In anycase it looks as though it is trivial to generate potential beta algebras using infinite ordinals, though he doesn't appear to have noticed this.