Geodesic
Оn the congruence lattices of periodic unary algebras
Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2014), pp. 27-30.

Voir la notice de l'article provenant de la source Math-Net.Ru

The author describes all commutative unary algebras with finite number of unary operations which have distributive lattice of congruences and cyclic elements in every operation. It proves the following result: Theorem 2. Let ${\mathbf A}$=${\langle A, f_1, f_2, \ldots, f_m \rangle }$ is a connected commutative unary algebra, $m\geq 1$ and $n_1, n_2, \ldots,n_m\geq 1$ — such a natural numbers, that $f_i^{n_i}(x)=x$ for every $i\leq m$ and every $x\in A$. Then the following condition are equivalent: (1) The lattice of congruence on ${\mathbf A}$ has a distributive property. (2) One can find natural numbers $k_1, k_2, \ldots, k_m\geq 1$ and such an unary operation $h$ on ${\mathbf A}$, that for every $i=1, 2, \ldots, m$ and every $x\in A$ it holds $f_i(x)=h^{k_i}(x)$.
Keywords: unary operation, commutative unary algebra, lattice of congruence, distributive property, cyclic element. 
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V. V. Popov. Оn the congruence lattices of periodic unary algebras. Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2014), pp. 27-30. http://geodesic.mathdoc.fr/item/VVGUM_2014_2_a3/

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