On Thursday, November 13, 2014 at 8:05:33 AM UTC-6, Jonathan Cender
wrote:

An additive identity element defined both in terms of subtraction and division.

In any group e/e = e has only one solution, where a/b is defined by a
b^{-1}. The same for e\e = e where a\b is defined by a^{-1} b.

Therefore the identity is defined entirely in terms of the inverse
operator -- and in a almost trivially simple way.

A group, itself, is characterized in term of its inverse operator by
the axioms: a/a = e, a/e = a and (a/c)/(b/c) = a/b -- which is ironic
since these are also the key properties that appear in the context of
quotient groups.

In additive notation this becomes a - a = 0, a - 0 = a and (a - c) - (b
- c) = a - b. And you can go even further: an Abelian group needs only
two identities: a - (b - c) = c - (b - a) and a - (a - b) = b, in
addition to an existential axiom for the identity (0 - 0 = 0;
equivalently: that everything is expressible as a difference). The
precondition (a - a = b - b) is a theorem of the first two axioms.

You can go further still! A TORSOR (i.e. a "group" in which all members
have equal standing) can be defined as an algebra with a single ternary operator a - b + c that satisfies the axioms: a - a + b = b = b - a + a
and (a - b + c) - d + e = a - b + (c - d + e). It is Abelian if a - b +
c = c - b + a.

These possess group bundle structures. The "fibre" T_o associated with
any element o of a torsor T is the group possessing operations defined
by a + b = a - o + b, a^{-1} = o - a + o. Each fibre group is provably isomorphic to the others and to the "standard group" dT that is
constructed by taking T x T modulo the identity (a, b - (c - d)) = (a -
(d - c), b). The equivalence classes a - b def= [(a,b)] give you the
operation of the group dT (and the fact that it *is* a group follows
from the torsor identities).

But wait (now in infomercial mode) ... there's more! An Affine Geometry
can also be defined in a similar way: a ternary operation [a,r,b] in A
for any elements a, b of a affine geometry A over a field F and element
r of the field F. The axioms here are: [a,0,b] = a, [a,1,b] = b and [a,rt(1-t),[b,s,c]] = [[a,rt(1-s),b],t,[a,rs(1-t),c]]. This corresponds
to the operation [a,r,b] = (1 - r)a + rb. The affine operation a - b +
c is recouped from the identity a - b + c = [b,1/(1-r),a],r,[b,1/r,c]]
valid for any r not equal to 0 or 1.

These axioms produce affine geoetries unless the field is the 2 or 3
element field. The axioms are a bit too weak when the field F has 3
elements (and produce the commutative form of a structure known as a
quandle, with the operation ab = [a,-1,b]). Affine geoetries over
2-element fields require their own treatment -- I think these are the
same thing as Boolean rings.

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