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

It is a standard device in mathematicsto interpret a multi-variate functionas a nesting of functions of onevariable.  So, for example,F(x,y) = (F(x))(y)Curry attributes this method to Schoenfinkeland uses it to interpret concatenationsof strings.  So, for example,(F(x))(y) = Fxyand, in general,((...(((a(b))(c))(d))...)(x)) = abcd...xUsing the complete connectives, it ispossible to introduce a binary producton the domain of the connectivityalgebra based on this idea.Consider that if the interpretationof a set of parenthesis is taken as specifyinginterpretation 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 connectivityalgebra, this nested product will resolveto some element of the domain.  By virtueof the configuration of parentheseslet this operation be called multiplicationon the right.For multiplication on the left, one wouldhave 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 ofthe domain.However, one is now confronted withthe problem that a simple set ofparentheses is once again ambiguous(x,y)Because the multiplication is notnecessarily commutative, one mustnow speak of a concatenationas interpreting well-formedexpressions.The nested sequences that distinguish thetwo operations have as their innermostforms either(x(y))for multiplication on the right or((x)y)for multiplication on the leftIf one represents that ambiguous stateusing only parentheses, then(()())motivates a definition of well-formednesssufficient to identify whether or not the firstoperation is right multiplication orleft multiplication.Let a multiplication expression be takenas well-formed if the expression takes(())as its atom.  Given this, a concatenationxyinterprets the expression((())())as(x(y))or, multiplication on the rightand, similarly, interprets theexpression(()(()))as((x)y)or, multiplication on the leftAnd, no expression is well-formedit an atom occurs in any other position.Note that(x(x)) = ((x)x)and takes any given element intothat element which corresponds withits image under the collineationNegation:axis-THISline elements-THIS SOME OTHERS NO ALLThus, 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 thenew multiplication product is not an intensionalfunction.
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