Popov V.V. Оn the congruence lattices of periodic unary algebras
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Popov Vladimir Valеntinovich
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Computer Science and Experimental Mathematics Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
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 A = <A,f1,f2,...,fm> is a connected commutative unary algebra, m ≥ 1 and n1,n2,...,nm ≥ 1 — such a natural numbers, that fini(x) = x for every i ≤ m and every x ∈ A. Then the following condition are equivalent:
(1) The lattice of congruence on A has a distributive property.
(2) One can find natural numbers k1,k2,...,km ≥ 1 and such an unary operation ℎ on A, that for every i = 1,2,...,m and every x ∈ A it holds fi(x) = hki(x).
Theorem 2. Let A = <A,f1,f2,...,fm> is a connected commutative unary algebra, m ≥ 1 and n1,n2,...,nm ≥ 1 — such a natural numbers, that fini(x) = x for every i ≤ m and every x ∈ A. Then the following condition are equivalent:
(1) The lattice of congruence on A has a distributive property.
(2) One can find natural numbers k1,k2,...,km ≥ 1 and such an unary operation ℎ on A, that for every i = 1,2,...,m and every x ∈ A it holds fi(x) = hki(x).
Key words: unary operation, commutative unary algebra, lattice of congruence, distributive property, cyclic element.
Оn the Congruence Lattices of Periodic Unary Algebras by Popov V.V. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №2 (21) 2014 pp. 27-30
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