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Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, Issue 2(21), Pages 27–30
(Mi vvgum43)
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This article is cited in 1 scientific paper (total in 1 paper)
Mathematics
Оn the congruence lattices of periodic unary algebras
V. V. Popov Volgograd State University
Abstract:
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.
Citation:
V. V. Popov, “Оn the congruence lattices of periodic unary algebras”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, no. 2(21), 27–30
Linking options:
https://www.mathnet.ru/eng/vvgum43 https://www.mathnet.ru/eng/vvgum/y2014/i2/p27
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