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. (1+>"a" | 3+>"c") is a super string.

2. For finite_Sets A, super(A)::= Nat<>->A
3. For A, elements(A)::=Nat><A.
4. For x: super(A), end(x)::Nat= if(x={}, 0, max(pre(x)) ).
5. For x,y:super(A), x!y::@(Nat,A)= x | y(_+end(x)).
   
6. (above)|-MONOID( super(A), (!), {}).
7. |()- (monoid) #A==>super(A).
8. |()- super(A) generated_by elements(A).
   
9. (above)|-Unique factorisation!

10. 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
    

11. |-For P,Q:@super(X), x!P, P!x, P!Q are defined and in @super(A).

    We have a typical induction schema:

12. SUPER_INDUCTION::=
    Net{

    1. P::@super(A).
    2. {} in P.
    3. For all e:elements(A), a!P ==>P

       
       

    4. (above)|-P=super(A).

    
    }=::SUPER_INDUCTION.
    
    
13. |-SUPER_INDUCTION.

    We can reduce a superstring to just the elements in a subset of the natural numbers:

14. For N:@Nat, x:super(A), N!x::=N;x.
    
15. (above)|-N!x={ i+>a || i:N and (i+>a) in x }.

. . . . . . . . . ( end of section SuperStrings) <<Contents | End>>

HyperStrings

Replace Nat by Positive Real,
1. hyper(A)::= { x: Real<>->A || one lub(x) and one glb(x) },
2. end(x)::=lub(pre(x)), ...

. . . . . . . . . ( end of section HyperStrings) <<Contents | End>>

histories

1. histories(A)::= { x: Real<>->bag(A) || one lub(x) and one glb(x) },
2. end(x)::=lub(pre(x)), ...
3. For x,y:histories(A), x!y::@(Real,bag(A))= { (t,e) || t in pre(x)|pre(y) and e=x(t)+y(t) }.

. . . . . . . . . ( end of section histories) <<Contents | End>>

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.

. . . . . . . . . ( end of section Super-general) <<Contents | End>>

Notes on MATHS Notation

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle = Net{radius:Positive Real, center:Point}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

Notes on the Underlying Logic of MATHS

For a more rigorous description of the standard notations see