Оn the congruence lattices of periodic unary algebras

V. V. Popov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2014), pp. 27-30

Résumé

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 $f_i(x)=h^{k_i}(x)$.

Détail

Keywords: unary operation, commutative unary algebra, lattice of congruence, distributive property, cyclic element.