A Calculus of Number Based on Spatial Forms - University of ...

34
A pro<strong>of</strong> <strong>of</strong> inverse cancellation also follows from inversion:
=A=A:
Besides collecting and cancelling, the inverse operation can be \promoted" over
the modier form, (A[:::]):The theorem <strong>of</strong> inverse promotion derives this property.
Theorem 3 (Inverse Promotion) An inverse boundary can be promoted over the
composite boundary, (A[:::]):
=(A[]):
Pro<strong>of</strong>. For all A; B; C 2 B;B 6= 2;
Given
(A[ ]) Void Multiplication
(A[B]) Inversion
(A[B])(A[]) Distribution
(A[]) Inverse
These three theorems appear in standard and boundary numbers as:
(,x)+(,y)=,(x+y) versus =
,(,x) =x versus =x
,(xy) =x(,y) versus =(x[])
5.3.2 Integers
Since the boundary calculus now includes the additive inverse, natural numbers can
be extended to the integers. The instance boundary serves as a unit for the natural
numbers and the inverse boundary inverts that unit. The negative unit is given by
: These numbers add by spatial collection and multiply by the form using the
abstract boundary.
The void and elements ;;;;;;::: compose the set I <strong>of</strong> integers
within the boundary calculus. The successor function is S(A) ! A and the predecessor
function is P (A) ! A: From the void, successors to the void take the form
::: and predecessors to the void take the form : The predecessor function
gives ; which equals byinverse collection.