David Hobby's question about absolute nonassociativity prompts me to ask
the following question that's been on my mind lately about some
nonassociative operations that are themselves closely related to David's
example of material implication. If my question has already been
considered previously pointers would be welcome.

Quite apart from the notations for join (x+y vs. xvy) and meet (xy vs.
x^y), the notations for relational composition (the monoid
multiplication, as x;y or xoy or x.y), the unit (1' vs. I vs. e vs. 1
(linear logic)), and the two residuals (x\y vs. x->y vs. x-oy, x/y vs.
x<-y vs. xo-y) have varied all over the place. The least variation
seems to be for conjugation as De Morgan dual to residuation, namely
x |> y and x <| y using the triangles &#9655; and &#9665;, as used by
Jónsson and Tsinakis (AU 30 469-478 1993). (Have there been any other
notations for the conjugates?) The De Morgan duality is with respect to
a fixed "denominator" y, as in x/y = (x' <| y)', x <| y = (x'/y)'.

If you were relocated to another planet and could start all over again,
what would you pick for the residuals and the conjugates?

Looking outside residuated lattice theory for notational inspiration,
logical implication and numerical division are both contravariant in one
argument: the antecedent in the case of implication, the denominator in
the case of division. If one were considering just residuation and not
conjugation, and division was the first mixed-variance operation to come
to mind, it would be very natural to associate residuation to division
and notate it as x/y (for the left residual), the choice made by Ward
and Dilworth in 1939.

But what if they had considered residuation and conjugation as a
package? Would they think of implication as the second mixed-variance
binary operation? And if so how would they assign division and implication?

Modus ponens is the rule

x, x->y |- y.

Identifying |- with <= and the comma with the monoid multiplication
(as opposed to meet), modus ponens expresses a property of residuation,
not of conjugation.

With numerical division on the other hand we have the unary reciprocal
operation 1/x, an involution for nonzero x. Writing I for the monoid
unit, the defining characteristic of a relation algebra distinguishing
it from other residuated Boolean algebras is that I <| x as a function
of x is an involution (true if and only if x |> I is an involution),
namely converse. Freyd in his notion of allegory even goes so far as to
call reciprocal what the rest of us are content to call converse.
Residuation does not yield an involution in so direct a way.

Planning ahead for my own alien abduction, my inclination would on the
above grounds be to write the left and right residuals as
x o- y and x -o y respectively (since -> is already in use for
material implication), and their respective De Morgan duals, the left
and right conjugates, as x/y and x\y respectively. With I as the monoid
unit (to avoid the common use of 1 as top, in agreement with the
corresponding conventions for monoidal categories), the reciprocal of x
is I/x, satisfying I/(I/x) = x as one would expect.

Are there just as compelling technical arguments for the other way
round? For that matter are there even more suitable nonassociative
mixed-variance operations drawn from outside relation algebra? (Bra and
ket come to mind, but they are so clumsy in practice that I find it
amazing physicists continue to use them instead of a tidy infix operator.)

Incidentally this is all apropos of the relative connectives. This
logical/numerical dichotomy collapses for material implication, for
which (x -> 0) -> 0 = x, i.e. ->0 is an involution, namely complement.
The De Morgan dual of x->, namely (x -> y')', collapses to x^y
(conjunction). But now x^1 = x, whence (x^1)^1 = x, making ^1 an
involution too, boringly so however being the identity.

Vaughan Pratt