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.