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Subassociative groupoids. (English) Zbl 1123.20059
Summary: When $$\langle G;\diamond\rangle$$ is a groupoid with binary operation $$\diamond\colon G^2\to G$$, and when $$k\in\mathbb{N}:=\{1,2,3,\dots,\}$$, then $$F^\sigma(k)$$ denotes the set of all formal products $$\mathbf u$$ on $$k$$ independent variables. It is well known that $$|F^\sigma(k)|=C(k)$$, where $$C(k)$$ is the $$k$$-th Catalan number.
Each word $$\mathbf u\in F^\sigma(k)$$ induces a function $$\mathbf u\colon G^k\to G$$ given by $$\mathbf u\colon\vec g\mapsto\mathbf u(\diamond,\vec g)$$, where $$\mathbf u(\diamond,\vec g)$$ is the interpretation in $$\langle G,\diamond\rangle$$ of $$\mathbf u$$ as a $$\diamond$$-product of the sequence $$\vec g:=\langle g_0,g_1,\dots,g_{k-1}\rangle\in G^k$$.
Write $$\mathbf u=_\diamond\mathbf v$$ for $$\{\mathbf{u,v}\}\subseteq F^\sigma(k)$$ iff $$\mathbf u(\diamond,\vec g)=\mathbf v(\diamond,\vec g)$$ whenever $$\vec g\in G^k$$. This $$=_\diamond$$ is an equivalence relation on the set $$F^\sigma:=\bigcup\{F^\sigma(k):k\in\mathbb{N}\}$$. The sequence $$\mathbf{SaT}(\langle G;\diamond\rangle):=\langle|F^\sigma(k)/=_\diamond|\rangle_{k=2}^\infty$$ presents the subassociativity types of $$\langle G;\diamond\rangle$$.
We calculate $$\mathbf{SaT}(G)$$ for a few evocative groupoids $$G:=\langle G;\diamond\rangle$$, and we initiate a study of the partitions $$F^\sigma(k)/=_\diamond$$. Each equivalence class of the completely free groupoid $$F^\sigma$$ is a singleton, and so $$F^\sigma$$ realizes the theoretical minimum $$k$$-associativity for each $$k\in\mathbb{N}$$. We propose for each $$k$$ a minimally $$k$$-associative class of finite groupoids.
MSC:
 20N02 Sets with a single binary operation (groupoids) 08A02 Relational systems, laws of composition 05A15 Exact enumeration problems, generating functions