M. S. Braitt and D. Silberger
Subassociative groupoids

When (G;o) is a groupoid with binary operation o:G²-->G, and when k from N={1,2,3,...}, then F^s(k) denotes the set of all formal products u on k independent variables. It is well known that |F^s(k)|=C_k, where C_k is the k-th Catalan number.

Each word u in F^s(k) induces a function u:G^k --> G given by u:w_g-->u(o,w_g), where u(o,w_g) is the interpretation in (G;o) of u as a o--product of the sequence w_g = (g₀,g₁,...,g_k-1) in G^k.

Write u=_ov for {u,v} contained in F^s(k) iff u(o,w_g)=v(o,w_g) whenever w_g in G^k. This =_o is an equivalence relation on the set F^s = U(F^s(k)) where k in N. The sequence SaT(G;o)= (|F^s(k)/=_o|) presents the subassociativity types of (G;o).

We calculate SaT(G) for a few evocative groupoids (G;o), and we initiate a study of the partitions F^s(k)=_o. Each equivalence class of the completely free groupoid F^s is a singleton, and so F^s realizes the theoretical minimum k--associativity for each k from N. We propose for each k a minimally k--associative class of finite groupoids.

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