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Zapiski Nauchnykh Seminarov POMI, 2001, Volume 279, Pages 24–60 (Mi znsl1452)  

This article is cited in 2 scientific papers (total in 2 papers)

Metrized semigroups

V. N. Berestovskii, V. M. Gichev

Omsk State University
Full-text PDF (356 kB) Citations (2)
Abstract: The notion metrized order (antimetric) on a topological group is characterized by three equivalent systems of axioms and connected with pointed locally generated semigroups. In the present paper, these notions are discussed and new results are announced; the main result is an analog of the following fact in metric geometry: every left-invariant inner metric on a Lie group is Finsler (maybe, nonholonomic). In the situation considered, a norm is replaced by an antinorm, and a metric by an antimetric. Examples are given, showing the complexity of these structures and their prevalence. Among them are: a nonholonomic antimetric on Heisenberg group, an antimetric on a nonnilpotent group admitting dilatations, a pointed locally generated semigroup in the Hilbert space with trivial tangent cone, antinorms connected with the Brunn–Minkowski inequality and the Shannon entorpy, a discontinuous antinorm on a Lie algebra defining a continuous antimetric on the Lie group, and an example of the converse situation. Several problems are formulated.
Received: 25.12.2000
English version:
Journal of Mathematical Sciences (New York), 2004, Volume 119, Issue 1, Pages 10–29
DOI: https://doi.org/10.1023/B:JOTH.0000008737.57612.5d
Bibliographic databases:
UDC: 515.122.4
Language: Russian
Citation: V. N. Berestovskii, V. M. Gichev, “Metrized semigroups”, Geometry and topology. Part 6, Zap. Nauchn. Sem. POMI, 279, POMI, St. Petersburg, 2001, 24–60; J. Math. Sci. (N. Y.), 119:1 (2004), 10–29
Citation in format AMSBIB
\Bibitem{BerGic01}
\by V.~N.~Berestovskii, V.~M.~Gichev
\paper Metrized semigroups
\inbook Geometry and topology. Part~6
\serial Zap. Nauchn. Sem. POMI
\yr 2001
\vol 279
\pages 24--60
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1452}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1846071}
\zmath{https://zbmath.org/?q=an:1063.22007}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 119
\issue 1
\pages 10--29
\crossref{https://doi.org/10.1023/B:JOTH.0000008737.57612.5d}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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