Given an addition, we can look at automorphisms of it; we can construe repeated addition as natural scaling; each natural scaling is a automorphism and necessarily commutes with all automorphisms of the addition. A composite of scalings scales by the product of the scale factors of the composed scalings. We can extend scaling to some class of other automorphisms, among which we construe composition as a multiplication that distributes over our addition. This may include scaling by rationals, reals or even complex scalars. Among the automorphisms of the addition, we have an induced pointwise addition. If we can embed our addition in the addition among its automorphisms, we can interpret this thereby as a multiplication on the same values as our addition. If the addition and multiplication thus combined behave suitably, we may get a ringlet, with various properties, perhaps even an ordered field.
Just as the automorphisms of addition include natural scaling, automorphisms of a commutatative multiplication include natural powers; so we can think of automorphisms of a multiplication as at least analogous to powers. We could, of course, repeat the whole structure above to another layer atop multiplication, as it arises atop addition; however, we could just as readilly simply set aside our original addition and re-express the combiner we initially thought of as multiplication as our new addition, on which we build the same structure as above. This can alternatively be phrased in terms of an isomorphism from our original multiplication to an addition; the isomorphism lets us use this addition as a model of our multiplication, apply the usual reasonings of arithmetic to this second addition and thereby, back through the isomorhism, infer properties of our original multiplication. So I now turn to homomorphisms between addition and multiplication, with particular attention to where it's from a ringlet's addition to a restriction of that ringlet's multiplication, particularly when addition of the latter's values defines a full order on the ringlet.
In particular, an important feature of such homomorphisms is how fast they grow as a function of their inputs, which I shall compare with one another, with scalings, powers and with the factorial function (: n! ←n :) defined by n! = product(: n−i ←i |n), so 0! = 1 and (1+n)! = (1+n).n! for each natural n. From this I also derive the combinatoric function F(i, j) = (i+j)!/i!/j!, which gives (among other things) the number of ways of partitioning i+j things into i things separate from the other j things.
Since we have either the naturals or the natural multiples of the input
ringlet's multiplicative identity among our inputs, I'll characterise speed of
growth in terms of the outputs to which each function maps the naturals. I'll
characterise outputs as bigger
or smaller
using a tacit ordering
on output values, although this may be a magnitude
ordering (as among the
complex numbers) rather than a full order (as among the reals). In particular,
big negative values (the additive inverses of big positive values) are here
presumed to be bigger than
small positive values.
On any ringlet, each power(n) is a mapping from the ringlet's values to the ringlet's values; we can multiply such mappings pointwise by other such mappings; when we do, power(n).power(m) = power(n+m) and power(n)&on;power(m) = power(n.m); so we can construe power as a homomorphism from natural addition to the multiplication the ringlet induces on such mappings and, at the same time, as a homomorphism from multiplication among {naturals} to composition among such functions. When the ringlet's multiplication is abelian, the natural powers are automorphisms of the ringlet's multiplication. In some cases, power can also be extended to a homomorphism from rational or even real addition to a ringlet's multiplication.
for value x of our ringlet, transpose(power, x) = (: power(n, x) ←n
:{naturals}) is a homomorphism from natural addition (of which the ringlet's
addition contains an image, its additon's retriction to its natural scalings of
its multiplicative identity) to the ringlet's multiplication. In power(n, x), n
is referred to as the exponent; so a function that varies as an output of
transpose(power) does, i.e. a homomorphism from an addition to a multiplication,
is described as an exponential
. In particular, power
itself is an exponential.
Any exponential e gives us e(x +y) = e(x).e(y); it thus transforms repeated
addition into repeated multiplication so, for natural n, e(n.x) = power(n,
e(x)). When e maps from a ringlet's values, we have an image of {naturals} in
the ringlet, for which e(n) = power(n, e(1)). When e is from a ringlet's
addition to the same ringlet's multiplication, we can compare e to any output of
power. When the ringlet is ordered, with e's outputs all positive (as e(2.x) =
power(2, e(x)) necessarily is for ever x, at least), we can compare e(1) to 1;
if e(1) < 1 then successive powers of e(1) shall diminish towards zero; if
e(1) is 1, then e maps all naturals to 1; otherwise, e(1) > and e is
described as an increasing exponential
, with e(n) < e(n+1) for each
natural n.
For an increasing exponential, 1 = e(0) < e(1) < … < e(n) < e(n+1), increasing at each step by the factor e(1). As long as we can write e(1) = 1 +k for some k > 0, we have e(n) = power(n, 1 +k) which the binomial formula expresses for us as sum(: power(1, i).power(k, j).F(i, j) ← i+j = n :) = sum(: power(k, j).n!/j!/(n −j)! ←j :1+n) in which, as k > 0, each term is positive. Its first two are power(0, k).n!/0!/n! = 1 and power(1, k).n!/1!/(n−1)! = n.k, which we usually presume shall be > 1 for at least some some n, no matter how small k may be. (If your ringlet includes inifinitessimals, indeed, we need to group e(1) = 1 + infinitessimal with the case e(1) = 1, precisely because this presumption fails.) That gives us e(n) > 1 +n.k > 2; as long as we have at least some n for which e(n) > 2, we have e(h.n) > power(h, 2) for each natural h.
Written by Eddy.