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 Algebra Logika, 2012, Volume 51, Number 5, Pages 623–637 (Mi al554)

Automorphisms of Boolean algebras definable by fixed elements

D. E. Pal'chunovab, A. V. Trofimova

a Novosibirsk State University, Novosibirsk, Russia
b Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia

Abstract: Enriched Boolean algebras are studied. We give an answer to the question asking under which conditions, given a subalgebra of a Boolean algebra, we can uniquely reconstruct an automorphism for which the given subalgebra is a subalgebra of fixed elements. Also we furnish a complete description of subalgebras of Boolean algebras that are fixed subalgebras of automorphisms definable by fixed elements. It is proved that an automorphism of a Boolean algebra is defined by fixed elements iff it is an involution. Subalgebras of fixed elements of automorphisms of atomic and superatomic Boolean algebras are examined. It is shown that an automorphism of a distributive lattice is defined by fixed elements iff it is an involution, and that this is untrue of finite modular lattices.

Keywords: Boolean algebra, automorphism, Boolean algebras with distinguished subalgebra, fixed elements of automorphism, involution, distributive lattice.

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English version:
Algebra and Logic, 2012, 51:5, 415–424

Bibliographic databases:

UDC: 512.563
Revised: 24.04.2012

Citation: D. E. Pal'chunov, A. V. Trofimov, “Automorphisms of Boolean algebras definable by fixed elements”, Algebra Logika, 51:5 (2012), 623–637; Algebra and Logic, 51:5 (2012), 415–424

Citation in format AMSBIB
\Bibitem{PalTro12} \by D.~E.~Pal'chunov, A.~V.~Trofimov \paper Automorphisms of Boolean algebras definable by fixed elements \jour Algebra Logika \yr 2012 \vol 51 \issue 5 \pages 623--637 \mathnet{http://mi.mathnet.ru/al554} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3075103} \zmath{https://zbmath.org/?q=an:06138171} \transl \jour Algebra and Logic \yr 2012 \vol 51 \issue 5 \pages 415--424 \crossref{https://doi.org/10.1007/s10469-012-9201-x} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000312401000004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84871395041} 

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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. N. A. Bazhenov, “D.c.e. degrees of categoricity for Boolean algebras with a distinguished automorphism”, J. Math. Sci., 211:6 (2015), 738–746
2. D. E. Pal'chunov, A. V. Trofimov, “Finitely-axiomatizable superatomic Boolean algebras with distinguished dense subalgebra of finite width”, Siberian Math. J., 57:6 (2016), 1066–1076
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