ONT Re: Relations And Their Divisitudes
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RATD. Note 6
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RC = Richard Cooper
RC: The definition of a dyadic relation is that
it takes two arguments and produces a result.
So for example, "+" and "-" are dyadic:
A + B -> C
A - B -> D
No, you are confusing functions, or operations, with relations.
The way that this is commonly said in math courses, where they
use "binary" and "ternary" instead of "dyadic" and "triadic",
is like this: "A binary operation is a ternary relation".
To translate: "A 2-adic operation is a 3-adic relation".
In these terms, the binary operation that is denoted by "+",
let's say on the integers J, is a function + : J x J -> J.
Any function of the type f : X x Y -> Z is a special sort
of 3-adic relation, which we may write in a polymorphous
fashion as f c X x Y x Z.
The formal structure of interest in the case of integer addition
is the mathematical group G = (J, +) c J x J x J, which we may
present in various ways, for example, taking <x, y, z> in J^3
to be in the group G if and only if x = y + z, in which case
we call x the "sum" of y and z. Expressed in an alternative
way, the rheme "___ is the sum of ___ and ___" is the frame
of a 3-adic relation.
RC: And the same is true of Cons:
(Cons A.B) -> C
(Car (A.B) -> A
(Cdr (A.B) -> (B)
And the same is true of Cons.
RC: Since "+" and "-" have historically been
among the dyadic functions, Cons must
also be dyadic, not triadic.
I can only recommend a review of the history.
RC: Thinking in relational terms, it is true that "+", "-"
and Cons can be written as triples, but that doesn't make
them triadic, at least not in the terminology with which
I am familiar.
RC: From compiler theory, the mondadic and dyadic functions
can be written in linear combinations such as:
f(x, y) = -sqrt((17*x+22*y-3)/7)
showing that "+", "-", "*" and "/" are considered the
usual dyadic relations, while "negate" and "sqrt"
are prefix unary operators.
RC: Any dyadic function can be represented as a table of triples --
its two arguments and its result -- but that doesn't make a
dyadic operator triadic.
It is necessary to distinguish operators,
which are functions, from the more general
class of relations.
Jon Awbrey
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