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Congruences of \(n\)-group and of associated Hosszú-Gluskin algebras. (English) Zbl 1030.20504

It is well known that an \(n\)-group \((Q;A)\) can be represented as \(A(x_1,\dots,x_n)=x_1\cdots x_n\) for some (binary) group \((R;\cdot)\), (\(Q\subseteq R\)) and by \(A(x_1,\dots,x_n)=x_1\cdot\varphi(x_2)\cdots\varphi^{n-1}(x_n)\cdot b\) where \((Q;A)\) is a (binary) group, \(\varphi\) is an automorphism and \(b\in Q\) such that \(\varphi(b)=b\) and \(\varphi^{n-1}(x)=bxb^{-1}\) (\(n\)-ary Hosszú-Gluskin algebra, shortly \(n\)HG algebra) [M. Hosszú, Publ. Math. 10, 88-92 (1964; Zbl 0118.26402)].
J. D. Monk and F. M. Sioson [in Fundam. Math. 72, 233-244 (1971; Zbl 0226.20079)] described the congruences of \((Q;A)\). The author uses the \(n\)HG algebra for the same purpose and shows that the congruence lattice of \((Q;A)\) is a sublattice of the congruence lattice of \((Q;\cdot)\) isomorphic to the lattice of all normal subgroups \((H;\cdot)\) of \((Q;\cdot)\) for which \(\varphi(H)=H\).

MSC:

20N15 \(n\)-ary systems \((n\ge 3)\)
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