Re: [Axiom-developer] algebras <=> groups

[Bertfried Fauser]

On Fri, 11 Jun 2004, root wrote:

Dear Tim,

> Axiom arranges algebras based on categorical properties such as
> associative, commutative, has one or multiple zeros, etc. The
> same is true, it appears, in group theory.
>
> It appears to me that there must be a group associated with every
> algebra with an isomorphism (the group rules have exact analogs
> in the algebra rules, the group elements have exact analogs in
> the algebra and vice-versa).
>
> (a) Is this true?
No,
> (b) Can you point me at a reference that details this?
Hence no pointer.

A group has the following AXIOMS:

0) G is a set with a selfaction GxG -> G on it
i) this action is associative  Gx (GxG) = (GxG) xG
ii) existence of a unit e, e xG =G = Gx e elementwise
iii) existence of a unique inverse  ()^-1 : G -> G such that
     h x g = e  and one calles h = g^(-1)


An algebra is build over a module (vector space)

A vectorspace is an abelian group of elements called vectors enriched by
the action of the scalars from R the vectors are build over. This abelian
group models the addition of an algebra. As a module the vector space is
identified with the algebra and denoted say A.

The multiplication of an algebra is a (possibly non associative) map
m : A x A -> A, so that this map is R-linear in both arguments
(Compatibility with teh addition, This definition is found (including the
nonassociativity, in the A1 Extension theory of Hermann Grassmann,
though he has not yet a word for the structure). In general an algebra
multiplication is not a group, since
i) m may be nonassociative -> no group analog, see theory of "loops"
ii) m may not have a unit at all, hence inverses cannot be defined
iiI) In almost all interesting cases A has not inverses for _any_ element
in A (check for a/the zero!)

The subset of lements which are multiplicatively invertible called the
group of units. R*=R\{0} as set. But zero is the unit element of the
additively writen group of addition and cannot be neglected.

Hence while a group has a unique and very well behaved connectivity, an
algebra has a much more interesting multiplication. Nevertheless, your
feeling that group and algebra structures are quite close together is not
so wrong. A categorial explanation how addition and multiplication are
connected (and are connected with the recursion, natural numbers, and
proof by induction) can be found in the recent Book of Lawvere/Rosenbrugh,
Sets for mathematics, Cambridge Univ Press, 2003.

> If we were to do group theory in Axiom it would be unbelievably
> sweet to arrange the group category structure side by side with
> the algebra category structure.
>
> That would allow the "lifting" trick

There is such a trick, as described in the book cited. Lets assume you
have a sucessor map suc which will be iterated to yield addition

        suc : x <- x+1
        add : x <- (suc^n)(x)

The multiplication can be understood as teh iteration of addition though

        mul : x <- (add^m)(n,x=0)

evaluated at zero. Here fun^n does mean fun(fun(fun(...)...), n-times.

This to implement in general would be marvelous, I am currently thinking
of vast generalizations of these structures, but doing symbolic algebra
with these structurs is bejond my ability.

ciao
BF.

% PD Dr Bertfried Fauser
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