.Open SuperStrings
Superstrings are a genralization of normal strings - all the same axioms but on a larger class of objects.
A superstring (unlike a normal string of symbols like "abc")
can hhave `holes` in it -- some subscripts are missing:
(1+>"a" | 3+>"c") is a super string.
For Sets A, finite A, super(A) ::= Nat<>->A
For A, elements(A) ::=Nat>super(A).
|()- super(A) generated_by elements(A).
()|-Unique factorisation!
For all x:super(A), x={} or for one e:element(A), y:super(A), x=e!y.
So define ... first(x)....rest(x)...last(x)...,
As always we have
|- For P,Q:@super(X), x!P, P!x, P!Q are defined and in @super(A).
We have a typical induction schema:
SUPER_INDUCTION::=Net{
P::@super(A).
{} in P.
For all e:elements(A), a!P ==>P
()|- P=super(A).
}=::SUPER_INDUCTION.
|- SUPER_INDUCTION.
We can reduce a superstring to just the elements in a subset of the
natural numbers:
For N:@Nat, x:super(A), N!x::=N;x.
()|-N!x={ i+>a || i:N and (i+>a) in x }.
.Close SuperStrings
.Open HyperStrings
Replace Nat by Positive Real,
hyper(A) ::= { x: Real<>->A || one lub(x) and one glb(x) },
end(x) ::=lub(pre(x)),
...
.Close HyperStrings
.Open histories
histories(A) ::= { x: Real<>->bag(A) || one lub(x) and one glb(x) },
end(x) ::=lub(pre(x)),
...
For x,y:histories(A), x!y::@(Real,bag(A))= { (t,e) || t in pre(x)|pre(y) and e=x(t)+y(t) }.
.Close histories
.Open Super-general
A is any set.
T is any set
S is a subset of @T such that each set t in S has a special maximum value
+ is an associative operation that preserves maximum values
the sum of two T's is greater than the max of them
S conatins all the finite subsets of T.
supergenstring=S<>->A.
.Close Super-general