Identities are simple enough as they are always true and are the tautologies of mathematics.

For instance 1=1 and x=x are both identities.

Mathematics uses identities all the time as consider the very easy solution of the following equation:

x-2 = 0

as you can use the identity 2=2 to find that

x-2+2 =2, so x=2

and you have a solution that IS so trivial that most probably don't think of identities when they solve such an equation.

A slightly more complicated but still easy use of identities is with completion of the square, as for instance, given

x

^{2}+ 2xy = z

^{2}

adding the identity y

^{2}= y

^{2}gives you

x

^{2}+ 2xy + y

^{2}= y

^{2}+ z

^{2}

which of course is just (x+y)

^{2}= y

^{2}+ z

^{2}, allowing you to take the square root of both sides and solve to find

x = -y +/- sqrt(y

^{2}+ z

^{2}).

Despite their importance in algebra where you cannot really do algebra without identities, development of the algebra of identities has not been extensive up until now which can be seen by comparing from my own research where I rely on more complicated identities still like

x

^{2}+ y

^{2}+ vz

^{2}= x

^{2}+ y

^{2}+ vz

^{2}

where to a large extent what I do is like what was shown before, except that the identities that I add end up being a lot more complicated, and are being used for more complex results.

That brings up the need for classification, where I've decided that the word "identity" doesn't do enough to cover everything so I introduced the phrase "tautological space", as with my use of identities the identity is not an afterthought at the end just to get to the answer, but a start of the argument and a dominant part of the proof.

Continuing now to classify all identities as tautological spaces, I've used the number of variables to give the dimension, so that with 4 variables you have a 4-dimensional tautological space.

The simplest identities like 2=2 that have no variables, have no dimension and I call them unary tautological spaces because they are all multiples of 1=1.

If any variables of the tautological space are raised to a power other than 1 then that is a hyperdimension and is shown by a set of natural numbers. Variables in a tautological space cannot be raised to a negative power, nor to 0 as that is redundant since instead a constant can be used.

For example with x

^{2}+ y

^{2}+ vz

^{2}= x

^{2}+ y

^{2}+ vz

^{2}you have a 4-dimensional tautological space with a hyperdimensional set of {2,2,1,2}.

Since the hyperdimensional set gives the number of dimensions, more compactly then you can describe a tautological space as follows.

T_S{2,2,1,2}

And that allows a complete classification of all possible identities.

Coming back to look this over I see that it does not, so I'm updating with a change in terminology.

The free variable which I traditionally call v is given by the 1, but I think I can use nesting to show where it is:

T_S(2,2,{1,2})

So that means a 4-dimensional tautological space where the conditional variables are all squared.

James Harris