Hi. We stumbled across the following results,
and wonder if anyone has already found them.
They certainly feel like things that could
have been done 50 to 100 years ago.

Consider groupoids, and look at "generalized
associative laws". These would be identities
similar to ((uv)w)((xy)z) = ((((uv)w)x)y)z.
(I'm using concatenation for the groupoid
operation.) Specifically, a generalized
associative law is an identity between two
terms, where the same variables each appear
once, in the same order, and the terms are
only distinguished by how they are parenthesized.

The result is that the following two element
groupoids satisfy no generalized associative
laws. The groupoid on {0,1} where the operation
is implication. (That is, 00 = 01 = 11 = 1, and
10 = 0.) And the groupoid on {0,1} where the
operation is NAND. (00 = 01 = 10 = 1, and 11 = 0).

Help?

---David