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 Diskr. Mat., 2009, Volume 21, Issue 3, Pages 119–131 (Mi dm1065)

On finite lattices of topologies of commutative unary algebras

A. V. Kartashova

Abstract: In this paper, we show that the lattice of topologies of a finite one-generated commutative unary algebra is isomorphic to the lattice of topologies of the characteristic semigroup of this algebra. With the use of this assertion, we give a characterisation of the class of all commutative unary algebras with linearly ordered lattices of topologies. It is proved that if the lattice of congruences or the lattice of topologies of a commutative unary algebra is finite, then the algebra itself is finite. Examples of infinite noncommutative unary algebras with finite lattices of topologies are given. It is proved that for an arbitrary functional signature containing at least one symbol with arity greater than 1 and for any integer $n\ge2$ there exists an infinite algebra of such signature whose lattice of topologies is linearly ordered and consists of $n$ elements.

DOI: https://doi.org/10.4213/dm1065

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English version:
Discrete Mathematics and Applications, 2009, 19:4, 431–443

Bibliographic databases:

UDC: 512.62
Revised: 13.12.2007

Citation: A. V. Kartashova, “On finite lattices of topologies of commutative unary algebras”, Diskr. Mat., 21:3 (2009), 119–131; Discrete Math. Appl., 19:4 (2009), 431–443

Citation in format AMSBIB
\Bibitem{Kar09} \by A.~V.~Kartashova \paper On finite lattices of topologies of commutative unary algebras \jour Diskr. Mat. \yr 2009 \vol 21 \issue 3 \pages 119--131 \mathnet{http://mi.mathnet.ru/dm1065} \crossref{https://doi.org/10.4213/dm1065} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2588657} \elib{http://elibrary.ru/item.asp?id=20730305} \transl \jour Discrete Math. Appl. \yr 2009 \vol 19 \issue 4 \pages 431--443 \crossref{https://doi.org/10.1515/DMA.2009.030} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70350493706} 

• http://mi.mathnet.ru/eng/dm1065
• https://doi.org/10.4213/dm1065
• http://mi.mathnet.ru/eng/dm/v21/i3/p119

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. K. Kartashov, “O nekotorykh rezultatakh i nereshennykh zadachakh teorii unarnykh algebr”, Chebyshevskii sb., 12:2 (2011), 18–26
2. A. V. Kartashova, “O reshetkakh kongruentsii i topologii unarnykh algebr”, Chebyshevskii sb., 12:2 (2011), 27–33
3. A. V. Kartashova, “O reshetkakh kongruentsii pryamykh summ silno svyaznykh kommutativnykh unarnykh algebr”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:4(2) (2013), 57–62
4. A. V. Kartashova, “On Commutative Unary Algebras with Totally Ordered Congruence Lattice”, Math. Notes, 95:1 (2014), 67–77
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