## 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 $$(n\ge 3)$$

### Keywords:

Hosszú-Gluskin algebras; lattices of congruences

### Citations:

Zbl 0118.26402; Zbl 0226.20079
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