Given an additive domain V, let AddOK(V) stand for {(V:f:(:V|)): f respects addition}. For any sub-collection, A, of AddOK(V), let Scaling(A) stand for {f in AddOK(V): for all a, b in A, f&on;a&on;b = a&on;b&on;f}, let Linear(A) stand for {g in AddOK(V): for all z in Scaling(A), z&on;g = g&on;z}, and let antiLinear(A) stand for {g&on;a: g in Linear(A), a in a}. I shall be particularly interested in the case where A = antiLinear(A) and every member of AddOK(V) is a sum whose terms are drawn only from Linear(A) and antiLinear(A).
The definitions immediately tell us:
and we may infer
since z&on;(g+f) = z&on;g + z&on;f = g&on;z + f&on;z = (g+f)&on;z whenever g and f are in Linear(A) and z is in Scaling(A)
since z&on;g = g&on;z and z&on;f = f&on;z do imply z&on;f&on;g = f&on;z&on;g = f&on;g&on;z
since f is g&on;b for some g in Linear(A) and b in A, whence f&on;a is g&on;b&on;a with g in Linear(A) and b, a in A, whence b&on;a in Linear(A); so f&on;a is a composite of members of Linear(A)
Given two members of antiLinear(A), can we expect their composite to be in Linear(A) ? It's of form g&on;a&on;f&on;b with g, f in Linear(A), a, b in A. Can we be sure a&on;f is in antiLinear(A), given a in A and f in Linear(A) ? If g&on;b is in Linear(A) for some b in antiLinear(A), can we infer that g is antiLinear ?
If A is closed under addition and subsumes {a&on;b&on;c: a, b, c in A},
I'll say that it presages
{(V:f:(:V|))
shadow(A) = {a, b&on;c, a+b&on;c: a, b, c in A}. This is equal to A if it is
closed under composition and addition; I am interested in it when A is closed
under addition but not composition (e.g. A = {antilinears}). Likewise, define
S(A) = {(V;f|(:V|)): f respects addition and commutes with all members of {b&on;c:
b, c in A}} and note that this always subsumes {real scalings}. Now, for any
z in S(A), I know z commutes with all members of {b&on;c: b, c in A}: when
I compose it with just one member of A, what happens ?
Given a collection, A, of mappings (V:|(:V|)) which respect addition, I'll
say that A shadows
{mapping (V:f|(:V|)): f respect addition} precisely if
this last is equal to {a, b&on;c, a + b&on;c: a, b, c in A}. The mappings in A
will be described as antilinear
, those in {b&on;c: b, c in A} as linear;
mappings which respect addition and commute with all linears will be described
as scalings. I shall need to show that, for every scaling z there is a scaling
s for which: a in A implies a&on;z = s&on;a; s and z will then be described as
conjugate
to one another. If A is closed under composition and addition,
it's equal to any collection it shadows.
To go from real scalings to (at least) complex ones, I need an abelian
family of mappings (V:|(:V|)), among which composition needs to be closed and
cancellable though possibly not complete, to construe as the non-zero scalings;
these shall include the positive real scalings, necessarily as a sub-family.
Linear maps are those which respect addition and commute with all these
scalings. We can then express our family as a product of two families; the
positive real scalings and a family of phases
; the inverse automorphism
of the latter, combined with the identity automorphism of the former, gives an
authomorphism of the scalings known as conjugation - preserve scale, invert
phase.
Antilinear maps are those which respect addition but when composed before a scaling yield the same result as composing after the conjugate of that scaling. I expect every mapping which respects addition to be expressible as a sum of terms, each of which is either linear or antilinear (i.e. as a linear plus an antilinear, at least when V has an additive identity to allow us the constant zero map, which is both linear and antilinear; absent this, we need to be able to leave out one kind of map, hence the wording given).
I may concievably want to allow a non-abelian family of phases, but at what
point does this degenerate into treating the rotations
of my additive
domain as scalar
multiplication ? Even in the abelian case, I'm not sure
I can define conjugation without somehow establishing a unit-sized
scaling for each phase, to do which without depending on conjugation I seem to
need the exp function (hence differentiation), though I may be able to play
games with the square root of -1, in so far as I can get my hands on it.
Does respect for addition enable me to define differentiation ? Not without multiplicative completeness (paddock-style, i.e. except for 0).
Written by Eddy.