Geodesic
On congruence lattices of direct sums of strongly connected commutative unary algebras
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 4, pp. 57-62 
A. V. Kartashova. On congruence lattices of direct sums of strongly connected commutative unary algebras. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 4, pp. 57-62. http://geodesic.mathdoc.fr/item/ISU_2013_13_4_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A union of mutually disjoint unary algebras is called their direct sum. A unary algebra is said to be strongly connected if it is generated by its arbitrary element. In the present paper we investigate congruence lattices of the class of all algebras with finitely many operations whose every connected component is strongly connected. We give a necessary and sufficient condition for an algebra from this class to have a distributive congruence lattice (Theorem 1). Besides, all distributive congruence lattices of algebras from the above class are discribed (Theorem 2).

[1] Berman J., “On the congruence lattices of unary algebras”, Proc. Amer. Math. Soc., 36:1 (1972), 34–38 | DOI | MR | Zbl

[2] Egorova D. P., Skornjakov L. A., “On congruence lattice of a unary algebra”, Ordered sets and lattices, 4, Saratov Univ. Press, Saratov, 1977, 28–40 | MR

[3] Egorova D. P., “The congruence lattice of a unary algebra”, Ordered sets and lattices, 5, Saratov Univ. Press, Saratov, 1978, 11–44 | MR

[4] Gratzer G., Shmidt E. T., “Characterizations of congruence lattices of abstract algebras”, Acta Sci. Math., 24 (1963), 34–59 | MR

[5] Johnson J., Seifert R. L., A survey of multi-unary algebras, Mimeographed seminar notes, U. C. Berkeley, N.Y., 1967, 16 pp.

[6] Esik Z., Imreh B., “Subdirectly irreducible commutative automata”, Acta Cybernetica, 5:3 (1981), 251–260 | MR | Zbl

[7] Kartashova A. V., “On finite lattices of topologies of commutative unary algebras”, Discrete Mathematics and Applications, 19:4 (2009), 431–443 | DOI | DOI | MR | Zbl

[8] Kartashov V. K., “Independent systems of elements in commutative unary algebras”, Algebra i teorija chisel: sovremennye problemy i prilozhenija, Tezisy dokl. mezhdunar. nauch. konf., Saratov, 2011, 29

[9] Akataev A. A., Smirnov D. M., “Lattices of submanifolds in manifolds of algebras”, Algebra and Logic, 7:1 (1968), 2–13 | DOI | MR | Zbl

[10] Kartashov V. K., “Lattices of quasivarieties of unars”, Siberian Mathematical Journal, 26:3 (1985), 346–357 | DOI | MR | Zbl | Zbl

[11] Boschenko A. P., Congruence lattices of unary algebras with two operations $f$ and $g$ which satisfy the identities $f(g(x))=g(f(x))=x$ or $f(g(x))= x$, Dep. VINITI 20.04.1998, No 1220–V98, Volgograd pedagogical university, Volgograd, 1998

[12] Mal'tsev A. I., Algebraic Systems, Springer-Verlag, Berlin, 1976, 392 pp. | MR | Zbl

[13] Esik Z., Imreh B., “Remarks on finite commutative automata”, Acta Cybernetica, 5:3 (1981), 143–146 | MR | Zbl

[14] Fuchs L., Abelian groups, Publ. House of the Hungar. Acad. Sci., Budapest, 1958, 367 pp. | MR