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This is an interesting question that arose a while back.  For a given monoid
M, let I(M) denote the monoid consisting of its finite subsets with the
operation: AB = { ab: a in A, b in B } and identity {1}.
Solve the fixed point equation: M = I(M).
Alternatively, let R(M) denote the monoid of rational subsets of M.  The
family of rational subsets is the smallest family containing the singletons
and empty set, closed under the operations:
           Product, defined as before;
           Union;
           Monoid closure: A* = {1} union A union AA union AAA union ...
Solve the fixed point equation: M = R(M).

-- I admit an additional a ternary operation [,,]
on Lie algebras (and similarly on affine Lie algebras)
which extends [,] since [x,y,0] = [x,y], see below.
The approach originates with the construction of a Lie 
theory for the (1+2)-superassociative law of Menger.
Especially interesting: [,,]-derivations are 
different from [,]-derivations. The only examples I know
satisfy [x,y,z] = [x,y,0] + [0,y,z] + [x,0,z] and are
derived from the underlying Lie algebra, see below.
My motivation is not mathematics but GUTs in physics.
Q1: Are these or similar results known? Please give references.
Q2: Are there any weird types of derivations in Lie theory?
Q3: Is there a Lie+ algebra which does not satisfy above identy?
    (more simply: Do I miss a trick to prove above identity?)
Please feel free to use any results but give credit.
-- In a few sections I sketch the additional operation [,,]:
#1 Of various failed attempts on a ternary operation for GUT 
   I sketch only J(x,y,z) since it influenced the outcome.
#2 Lie+ alggebra: Sketch of the additional operation [x,y,z] 
   and its connection with Lie algebras.
#3 Contains the connection with the superassociative law 
   of Menger and the notation of [,,]-derivation.
1 Motivation
The wellknown 7D crossproduct algebra derived from the
octonions has the 7-element projective plane PG(2,2) for
the structure constants and hence is named PG(2,2), too.
On its 8D completion with structure constants in 8-element 
affine space AG(3,2) [Oxley, Matroid theory, 1992, page 507]
it is possible to construct a multilinear ternary operation 
J which satisfies the nice distributive law 
J(J(d,e,f),g,h) = 
J(J(d,g,h),e,f) + J(d,J(e,g,h),f) + J(d,e,J(f,g,h))
provided the law 
J(a,a,b) = J(a,b,a) = J(b,a,a) = b is satisfied.
Unfortunately, the last identity is rather restrictive: 
If J is represented by a polynom of
an associative algebra then 2a=0. 
2 Lie algebra with ternary [,,] extending [,] 
The following construction works for any Lie ring
with commutation [x,y]. 
Define [x,y,z]:= [x,y] + [y,z] + [z,x]. Observe that
any element 'hugs' the other two elements. So I suggest
to call [x,y,z] the 'hug' of the elements x, y, z. Afterall,
it is a friendly new word not yet used in mathematics.
It is possible to recapture the original commutator since
[x,y,0] = [0,x,y] = [y,0,x] = [x,y]. The cost of this
'recapture of the commutator' is the loss of multilinearity:
(i) [w+ax,y,z] = [w,y,z] + [ax,y,z] - [0,y,z].
Observe that the last term is not 0 whereas the analog term
[0,y] of Lie theory is 0. Of course [0,0,z]= 0. 
(j) [x,y,z] = [y,z,x] = [z,x,y] = -[y,x,z]
Remember, that in Lie algebras the distributive law
[[x,y],z] = [[x,z],y] + [x,[y,z]] holds. It yields 
the following ternary distributive law for above [,,]:
(k) [[d,e,f],g,h] + [d,e,f] =
    [[d,g,h],e,f] + [d,[e,g,h],f] + [d,e,[f,g,h]] + [0,g,h]
Definition:
An additive group with ternary operation [,,] satisfying
(i) in any variable, (j) and (k) is called a 'Lie+ algebra'.
Remarks:
[x,y]:= [x,y,0] makes any 'Lie+ algebra' a Lie algebra.
Define ideals, simple algebras, ... as usual.
Applied to physics, simple 'Lie+ algebras' and representation
theory of 'Lie+ algebras' are just as in Lie machinery!
3 Mengers (1+2)-superassociative law --> 'Lie+ algebras'
-- Recall the basics of associative law and commutator:
The maps on a set S form a monoid which is a group if only
permutations are considered. If S is an additive commutative
monoid then the endomorphisms form a semiring; commutation
[f,g]:= fg-gf yields a Lie semiring which is a Lie ring if
the underlying additive monoid is a group.
-- Mengers generalization to maps on SxS:
For any three maps f,g,h: SxS --> SxS define a new map
: SxS --> SxS by
(x,y):= f(g(x,y),h(x,y)). Then the 'superassociative
law' <;G,H> = ,> holds.
Define the 'analogon of commutator' by [f,g,h]:= 
 +  +  +  +  + .
If S is an additive group and all maps are multilinear then
no further results seem to be possible unless you push on to
constructions of the type of the tensor product. 
But if all maps are 'joint' +-morphisms from the group
SxS into the group SxS then further results are possible:
Then the 'analogon of commutator' can be played back 
to associative algebras:
Observe that  =  +  where o is the 0-map.
Define f*g:=  and get an associative algebra with
commutation [f,g]:= f*g - g*f. 
Show that [f,g,h] = [f,g] + [g,h] + [h,f] holds.
I was surprised to end up within Lie machinery.
-- [,,]-Derivations:
The Jacoby identity is the blueprint for the definition
of derivation and yields immediately that any element 
any element 'a' induces a left derivation by  x-->[a,x] and 
a rigth derivation by  x-->[x,a], respectively. 
Yet the Jacoby identity is equivalent to a kind of 
distributive law as seen above. This suggests to define
a '[,,]-derivation' or short 'hug' in such a way that
for any pair of elements 'a' and 'b' the map 
x--> [x,a,b] is such a 'hug'. It could turn out that a nice
name for this map is UP(a,b) suggesting DOWN(a,b):= UP(b,a).
Yet I do not want to make any premature suggestions.
Definition:
A map H from a Lie+ algebra into a Lie+ algebra is a 'hug'
if it satisfies
(i) H(w+ax) = H(w) + H(ax) - H(0)
(k) H([x,y,z]) + [x,y,z]
    = [H(x),y,z] + [x,H(y,z] + [x,y,Hz] + H0
Theorem:
If H, I, J are hugs then [H,I,J] is again a hug where
[H,I,J](x):= [H(x),I(x),J(x)] is again.
Remark:
As in case of Lie algebras the proof of the Theorem depends
on the fact that for the maps H, I, J the identity
[H,I,J] = [H,I] + [I,J] + [J,H] holds with commutator 
taken with respect to composition of maps.

The subset of the unit circle consisting of the rational points:
        U_1_Q = { (cos (q pi), sin (q pi)): q a rational number }
should fit the description of the kind of space you're looking for.

Suppose I have some a source of unit vectors on $S^n\in R^{n+1}$, and
I want to test the hypothesis that these are uniformly distributed. Is
there some efficient way of doing this? There is an obvious way
involving summing spherical harmonics at the vectors in question, but
that seems computationally hard, and it is not quite clear how
confident should one be after doing the test for (say) $k$ harmonics.