For citation:
Kartashova A. V. On Congruence Lattices of Direct Sums of Strongly Connected Commutative Unary Algebras. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 4, pp. 57-62. DOI: 10.18500/1816-9791-2013-13-4-57-62
On Congruence Lattices of Direct Sums of Strongly Connected Commutative Unary Algebras
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).
- Berman J. On the congruence lattices of unary algebras. Proc. Amer. Math. Soc., 1972, vol. 36, no 1, pp. 34–38.
- Egorova D. P., Skornjakov L. A. O strukture kongrujencij unarnoj algebry [On congruence lattice of a unary algebra]. Uporjadochennye mnozhestva i reshetki [Ordered sets and lattices]. Saratov, Saratov Univ. Press, 1977, vol. 4, pp. 28–40 (in Russian).
- Egorova D. P. Struktura kongrujencij unarnoj algebry [The congruence lattice of a unary algebra]. Uporjadochennye mnozhestva i reshetki [Ordered sets and lattices]. Saratov, Saratov Univ. Press, 1978, vol. 5, pp. 11–44 (in Russian).
- Gratzer G., Shmidt E. T. Characterizations of congruence lattices of abstract algebras. Acta Sci. Math., 1963, vol. 24, pp. 34–59.
- Johnson J., Seifert R. L. A survey of multi-unary algebras. Mimeographed seminar notes. New York, U. C. Berkeley, 1967. 16 p.
- Esik Z., Imreh B. Subdirectly irreducible commutative automata // Acta Cybernetica, 1981, vol. 5, no 3, pp. 251–260.
- Kartashova A. V. On finite lattices of topologies of commutative unary algebras. Discrete Mathematics and Applications, 2009, vol. 19, iss. 4, pp. 431–443. DOI:10.1515/DMA.2009.030.
- Kartashov V. K. Nezavisimye sistemy jelementov v kommutativnyh unarnyh algebrah [Independent systems of elements in commutative unary algebras]. Algebra i teorija chisel: sovremennye problemy i prilozhenija :tezisy dokl. mezhdunar. nauch. konf., Saratov, 2011, pp. 29 (in Russian).
- Akataev A. A., Smirnov D. M. Lattices of submanifolds in manifolds of algebras. Algebra and Logic, 1968, vol. 7,iss. 1, pp.2–13. DOI: 10.1007/BF02218747.
- Kartashov V. K. Lattices of quasivarieties of unars. Siberian Mathematical Journal, 1985, vol. 26, iss. 3, pp. 346–357. DOI: 10.1007/BF00968621.
- Boschenko A. P. Reshetki kongrujencij unarnyh algebr s dvumja operacijami f i g, udovletvorjajushhimi tozhdestvam f(g(x)) = g(f(x)) = x ili f(g(x)) = x [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]. Volgograd pedagogical university. Volgograd, 1998. Dep. VINITI 20.04.1998, № 1220–В98 (in Russian).
- Mal’tsev A. I. Algebraic Systems. Berlin, Springer-Verlag, 1976, 392 p. (Rus. ed. : Mal’tsev A. I. Algebraicheskie sistemy. Moscow, Nauka, 1970, 392 p.)
- Esik Z., Imreh B. Remarks on finite commutative automata. Acta Cybernetica, 1981, vol. 5, iss. 3, pp. 143–146.
- Fuchs L. Abelian groups. Budapest, Publ. House of the Hungar. Acad. Sci., 1958, 367 p.
- 926 reads