Given a groupoid hG, ⋆i, and k ≥ 3, we say that G is antiassociative if an only if for all x1, x2, x3 ∈ G, (x1 ⋆ x2) ⋆ x3 and x1 ⋆ (x2 ⋆ x3) are never equal. Generalizing this, hG, ⋆i is k-antiassociative if and only if for all x1, x2, . . . , xk ∈ G, any two distinct expressions made by putting parentheses in x1 ⋆ x2 ⋆ x3 ⋆ . . . ⋆ xk are never equal. We prove that for every k ≥ 3, there exist finite groupoids that are k-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
In a groupoid, consider arbitrarily parenthesized expressions on the k variables x0, x1, . . . xk−1 where each xi appears once and all variables appear in order of their indices. We call these expressions k-ary formal products, and denote the set containing all of them by F σ (k). If u, v ∈ F σ (k) are distinct, the statement that u and v are equal for all values of x0, x1, . . . xk−1 is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds in them. These include the two groupoids on {0, 1} where the groupoid operation is implication and NAND, respectively.