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Given a groupoid G, ⋆ , and k ≥ 3, we say that G is antiassociative iff for all x 1

Given a groupoid G, ⋆, and k ≥ 3, we say that G is antias-sociative iff for all x 1

Consider arbitrarily parenthesized expressions on the k variables x 0 , x 1 , ..., x k−1 , where each x i appears exactly once and in the order of their indices. We call these expressions formal k-products. F σ (k) denotes the set of formal k-products. For {u, v} ⊆ F σ (k), the claim, that u and v produce equal elements in a groupoid G for all values assumed in G by the variables x i , attributes to G a generalized associative law. Many groupoids are completely dissociative; i.e., no generalized associative law holds for them; two examples are the groupoids on {0, 1} whose binary operations are implication and NAND. We prove a variety of results of that flavor. §1. Introduction.Our preceding paper, [2], begins an investigation of groupoids G := G; ⋆ in which the binary operation ⋆ : G × G → G fails to be associative; that is, those G for which there exists an ordered triple g 0 , g 1 , g 2 ∈ G 3 with (g 0 ⋆ g 1 ) ⋆ g 2 = g 0 ⋆ (g 1 ⋆ g 2 ). One task that finite G of this sort inspire is to specify how many of its |G| 3 distinct triples do associate. Indeed, [2] shows that, for every |G| ≥ 2, there exists G := G; ⋆ in which every triple fails to associate.The failure of some triples to associate induces ambiguity in the products of longer strings as well. Thus, whereas there are only two possibly distinct products of a triple of elements in G, there are 5 potentially distinct products of an ordered 4-tuple, 14 of a 5-tuple, 42 of a 6-tuple, etc. The

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