Polyadic analogs of the Cayley and Birkhoff theorems
Let \(A\) be an \(n\)-ary group with \(n\)-ary operation \(f\). On the set \(A^{n-1}\) define an \(n\)-ary operation \(g\) analogous with the \(n\)-ary operation, defined by Post for the \(n\)-ary transformations, by \[ \begin{multlined} g((a_1',\dots,a_{n-1}'),(a_1'',\dots,a_{n-1}''),\dots,(a_1^{(n)},\dots,a_{n-1}^{(n)}))=\\ =(f(a_1',a_2'',\dots,a_{n-1}^{(n)},a_1^{(n)}),f(a_2',\dots,a_{n-1}^{(n-2)},a_1^{(n-1)},a_2^{(n)}),\dots,f(a_{n-1}',a_1'',\dots,a_{n-1}^{(n)})).\end{multlined} \] The Cartesian product \(A^{n-1}\) endowed with the \(n\)-ary operation \(g\) is an \(n\)-ary group. The author establishes some results analogous with the Cayley and Birkhoff theorems. The main result is the following: For each \(n\)-ary group \((A,f)\) there exists an isomorphism of \((A^{n-1},g)\) with the \(n\)-ary group of all \(n\)-ary automorphisms of some sequence of universal algebras.
scientific article; zbMATH DE number 1752618