Date: Dec 7, 2012 5:27 AM
Author: fom
Subject: fom - 07 - connectivity algebra extension


It is a standard device in mathematics
to interpret a multi-variate function
as a nesting of functions of one
variable.  So, for example,

F(x,y) = (F(x))(y)


Curry attributes this method to Schoenfinkel
and uses it to interpret concatenations
of strings.  So, for example,

(F(x))(y) = Fxy

and, in general,

((...(((a(b))(c))(d))...)(x)) = abcd...x



Using the complete connectives, it is
possible to introduce a binary product
on the domain of the connectivity
algebra based on this idea.

Consider that if the interpretation
of a set of parenthesis is taken as specifying
interpretation by NOR, then


(x,y) = NOR(x,y) = (x(y))



and, more generally,



abcd =

(((a(b))(c))(d)) =

NOR(((a(b))(c)),d) =

NOR(NOR((a(b)),c),d) =

NOR(NOR(NOR(a,b),c),d)



Then, by the axioms of the connectivity
algebra, this nested product will resolve
to some element of the domain.  By virtue
of the configuration of parentheses
let this operation be called multiplication
on the right.


For multiplication on the left, one would
have a nested sequence appearing as


((d)((c)((b)a))) =

NAND(((a(b))(c)),d) =

NAND(NAND((a(b)),c),d) =

NAND(NAND(NAND(a,b),c),d)


which will also resolve to some element of
the domain.

However, one is now confronted with
the problem that a simple set of
parentheses is once again ambiguous

(x,y)

Because the multiplication is not
necessarily commutative, one must
now speak of a concatenation
as interpreting well-formed
expressions.

The nested sequences that distinguish the
two operations have as their innermost
forms either

(x(y))

for multiplication on the right or

((x)y)

for multiplication on the left

If one represents that ambiguous state
using only parentheses, then

(()())

motivates a definition of well-formedness
sufficient to identify whether or not the first
operation is right multiplication or
left multiplication.

Let a multiplication expression be taken
as well-formed if the expression takes

(())

as its atom.  Given this, a concatenation

xy

interprets the expression

((())())

as

(x(y))

or, multiplication on the right

and, similarly, interprets the

expression

(()(()))

as

((x)y)

or, multiplication on the left


And, no expression is well-formed
it an atom occurs in any other position.


Note that


(x(x)) = ((x)x)


and takes any given element into
that element which corresponds with
its image under the collineation


Negation:

axis-

THIS

line elements-

THIS SOME OTHERS NO ALL



Thus, the extended algebra has a signature with arities

<16, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2>

and, specification

<
{LEQ, OR, DENY, FLIP, NIF, NTRU, AND, NIMP, XOR, IMP, NAND, TRU, IF, 
FIX, LET, NOR},
LEQ,
OR,
DENY,
FLIP,
NIF,
NTRU,
AND,
NIMP,
XOR,
IMP,
NAND,
TRU,
IF,
FIX,
LET,
NOR,
*
 >


It must be given a specific listing now because the
new multiplication product is not an intensional
function.