There are three elements: x, y, z and a relation C:

x C y, y C z, z C x, x C x, y C y, z C z.

Let us introduce two binary operations with respect to the C: "the leftmost" (L) and "the rightmost" (R), i.e.

x L x = x L y = y L x = x, y L y = y L z = z L y = y, z L z = z L x = x L z = z

x R x = x R z = z R x = x, y R y = x R y = y R x = y, z R z = z R y = y R z = z.

Similar construction produces a multi-valued logic, if to use a linear order instead of the C, but this non-associative "logic" also has some applications. Yet, I failed to find any notes about that in a book about multi-valued logic. I would be glad to know, if described construction was used somewhere earlier to provide correct references in my works.