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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.
20N02 Sets with a single binary operation (groupoids)
08A02 Relational systems, laws of composition
05A15 Exact enumeration problems, generating functions