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)) = XY
and, more generally,
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
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
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
for multiplication on the right or
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
interprets the expression
or, multiplication on the right
and, similarly, interprets the
or, multiplication on the left
And, no expression is well-formed iF an atom occurs in any other position.
(x(x)) = ((x)x)
and takes any given element into that element which corresponds with its image under the collineation
NOT NO OTHER ALL SOME
Thus, the extended algebra has a signature with arities