Boundary mathematics and algebra of sets

(Corrected 23.12.08)

Brandon brought to my attention an interesting comparison between boundary mathematics and algebra of logic as developed in this blog. However I didn't quite follow his equivalences between the two systems. Of course this might be caused by the fact that I met with the subject of boundary mathematics first time in his blog. Nevertheless, the subject provoked me to think about similarities and differences between the two.

Jeffrey James, in his thesis "A Calculus of Number Based on Spatial Forms", bases his system on "making distinctions out of the void". The basic element is a single boundary in the void -- unit,  represented by o, or generally instance, represented by (X). The other boundaries are "black hole", represented by □, or generally abstract, represented by [X]; and inverse, represented by △ in empty case or generally by <X>. Only instance has actually a spatial interpretation as boundary. Boundaries can be collected and nested to form complex structures.

In following I will interpret [(X)] ≗ X as set X and formulate set algebra, taking lead from Wikipedia article. Translation into propositional formulas should be pretty straightforward by replacing set letters with propositional letters. I believe it may be regarded as extension of Jeffrey's notation, or as combination of his boundary mathematics with my algebra of logic. It's quite new to me, so there may be lot of things that need to be corrected.

Union of sets A and B: [AB]
Intersection of sets A and B: AB
Complement of set A (to universal set): <A>
Complement of set A relative to set B: <A>B
Difference of set A from B: A<B>
Symmetric difference of sets A and B: [(<A>B)(A<B>)]
Cartesian product of sets A and B: ([A][B])
Inclusion of set A in set B: [<A>B]

Commutativity: [AB] ≗ [BA]; AB ≗ BA
Associativity: [A[BC]] ≗ [[AB]C] ≗ [ABC]; A(BC) ≗ (AB)C ≗ ABC
Distributivity: [A(BC)] ≗ [AB][AC]; A[BC] ≗ [(AB)(AC)]

Null (or empty) set: □
Universal set: [o]?

Identity laws: [A□] ≗ A; A[o] ≗ A
Complement laws: [A<A>] ≗ [o]; A<A> ≗ □

Idempotent laws: [AA] ≗ A; AA ≗ A
Domination laws: [A[o]] ≗ [o]; A□ ≗ □
Absorption laws: [A(AB)] ≗ A; A[AB] ≗ A

Proof of idempotent laws:
[AA] ≗ [AA][o] ≗ [AA][A<A>] ≗ [A(A<A>)] ≗ [A□] ≗ A
AA ≗ [(AA)□] ≗ [(AA)(A<A>)] ≗ A[A<A>] ≗ A[o] ≗ A

DeMorgan's laws: <[AB]> ≗ <A><B>; <AB> ≗ [<A><B>]
Double complement or involution: <<A>> ≗ A
Complement laws of universal and empty sets: <[o]> ≗ □; <□> ≗ [o]

Uniqueness of complements: [<([AB]=[o] AB=□)> B=<A>] ≗ [<[AB]<(AB)>> B=<A>]

Reflexivity: [<A>A]
Antisymmetry: [<A>B][<B>A] ≗ A=B
Transitivity: [<[<A>B][<B>C]>[<A>C]]

If A, B and C are subsets of S:
Least element and greatest element: [<□>A]? and [<A>S]
Joins: [<A>[AB]]; [<[<A>C][<B>C]>[<[AB]>C]]
Meets: [<AB>A]; [<[<C>A][<C>B]>[<C>(AB)]]

[<A>B] ≗ AB=A ≗ [AB]=B ≗ A<B>=□ ≗ [<<B>><A>]

Relative complements:
C<AB> ≗ [(C<A>)(C<B>)]
C<[AB]> ≗ C<A><B>
C<B<A>> ≗ [(AC)(C<B>)]
[(B<A>)]C ≗ [([(BC)]<A>)] ≗ [(B[(C<A>)])]
[(B<A>)C] ≗ [BC]<A<C>>
A<A> ≗ □
□<A> ≗ □
A<□> ≗ A
B<A> ≗ <A>B
<B<A>> ≗ [A<B>]
[o]<A> ≗ <A>
A<[o]> ≗ □